How to find the sum of the perimeters of squares. Perimeter, area and volume

The ratio between the radius of a circle and the side length of a square. The distance from the center of the circumscribed circle to the top of the inscribed square is equal to the radius of the circle. To find the side of a square s, it is necessary to divide the square with a diagonal into 2 right triangles. Each of these triangles will have equal sides. a and b and the general hypotenuse with equal to twice the radius of the circumscribed circle ( 2r).

Use the Pythagorean theorem to find the side of a square. The Pythagorean theorem states that in any right-angled triangle with legs but and b and hypotenuse with: a 2 + b 2 = c 2... Since in our case but = b(don't forget we're looking at a square!) and we know that c = 2r, then we can rewrite and simplify this equation:

• a 2 + a 2 = (2r) 2 ""; now let's simplify this equation:
• 2a 2 = 4 (r) 2; now divide both sides of the equation by 2:
• (a 2) = 2 (r) 2; now extract Square root from both sides of the equation:
• a = √ (2r)... Thus, s = √ (2r).

1. Multiply the found side of the square by 4 to find its perimeter. In this case, the perimeter of the square is: P = 4√ (2r)... This formula can be rewritten like this: P = 4√2 * 4√r = 5.657r, where r is the radius of the circumscribed circle.

2. Example. Consider a square inscribed in a circle of radius 10. This means that the diagonal of the square is 2 * 10 = 20. Using the Pythagorean theorem, we get: 2 (a 2) = 20 2, i.e 2a 2 = 400. Now we divide both sides of the equation by 2 and get: a 2 = 200. Now let's take the square root of both sides of the equation and get: a = 14.142... Multiply this value by 4 and calculate the perimeter of the square: P = 56.57.

   • Note that you could have gotten the same result simply by multiplying the radius (10) by 5.657: 10 * 5,567 = 56,57 ; but this method is difficult to remember, so it is better to use the calculation process described above.

A square is a positive quadrangle (or a rhombus) in which all corners are right and the sides are equal. Like any other true polygon, square it is allowed to calculate perimeter and area. If the area square more famous, then discover his sides, and after that and perimeter will not be difficult.

Instructions

1. Square square is found by the formula: S = a? This means that in order to calculate the area square, it is necessary to multiply the lengths of its 2 sides by each other. As a consequence, if you know the area square, then when extracting a root from this value, it is possible to find out the length of the side square.Example: area square 36 cm?, In order to find out the side of this square, you need to extract the square root of the area value. Thus, the length of the side of a given square 6 cm

2. To find perimeter but square you need to add the lengths of all its sides. With the help of the formula, this can be expressed as follows: P = a + a + a + a. If we extract the root from the value of the area square, and then add the resulting value 4 times, then it is allowed to find perimeter square .

3. Example: Given a square with an area of ​​49 cm? You want to discover it perimeter Solution: First you need to extract the root of the square square:? 49 = 7 cm Then, calculating the length of the side square, it is allowed to calculate and perimeter: 7 + 7 + 7 + 7 = 28 cm Answer: perimeter square 49 cm? is 28 cm

Often in geometric problems it is required to find the length of the side of a square if its other parameters are known, such as area, diagonal or perimeter.

You will need

• Calculator

Instructions

1. If we know the area of ​​a square, then in order to find the side of the square, you need to extract the square root of the numerical value of the area (because the area of ​​the square is equal to the square of its side): a =? S, where a is the length of the side of the square; S is the area of ​​the square. The side of a square will be the linear unit of length that corresponds to the unit of area. Say, if the area of ​​a square is given in square centimeters, then the length of its side will turn out primitively in centimeters. Example: The area of ​​a square is 9 square meters. Find the length of the side of a square. Solution: a =? 9 = 3 Answer: The side of a square is 3 meters.

2. In the case when the perimeter of the square is known, to determine the length of the side it is necessary numerical value perimeter divided by four (because the square has four sides of the same length): a = P / 4, where: a - the length of the side of the square; P - the perimeter of the square The unit of the side of the square will be the same linear unit of length as y perimeter. Say, if the perimeter of a square is given in centimeters, then the length of its side is also obtained in centimeters. Example: The perimeter of a square is 20 meters. Find the length of a side of a square. Solution: a = 20/4 = 5 Answer: The length of a side of a square is 5 meters.

3. If the length of the diagonal of a square is famous, up to the length of its side will be equal to the length of its diagonal divided by the square root of 2 (by the Pythagorean theorem, because the adjacent sides of the square and the diagonal make up a right-angled isosceles triangle): a = d /? 2 (because . a ^ 2 + a ^ 2 = d ^ 2), where: a is the length of the side of the square; d is the length of the diagonal of the square. The unit of measurement for the side of the square will be the unit of measurement of length the same as for the diagonal. Say, if the diagonal of a square is measured in centimeters, then the length of its side will be in centimeters. Example: The diagonal of a square is 10 meters. Find the length of the side of the square. Solution: a = 10 /? 2, or approximately: 7.071 Answer: The length of the side of the square is equal to 10 /? 2, or approximately 1.071 meters.

The square is a lovely and simple flat geometric shape. It is a rectangle with equal sides. How to detect perimeter square, if the length of its side is famous?

Instructions

1. Before everyone, it is worth remembering that perimeter is nothing more than the sum of the lengths of the sides geometric shape... The square we are considering has four sides. Moreover, by definition square, all these sides are equal to each other. From these premises it follows simple formula to find perimeter but square – perimeter square equal to side length square multiplied by four: P = 4a, where a is the length of the side square .

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The perimeter is called the universal the length the boundaries of the figure are more often than each on the plane. A square is a positive quadrilateral, or a rhombus, in which all angles are straight, or a parallelogram, in which all sides and angles are equal.

You will need

• Knowledge of geometry.

Instructions

1. Perimeter square is equal to the sum the lengths of its sides. Because a square, in its essence, is a quadrangle, then it has four sides, which means that the perimeter is equal to the sum of the lengths of the four sides, or P = a + b + c + d.

2. A square, as can be seen from the definition, is a true geometric figure, which means that all its sides are equal. So a = b = c = d. Consequently, P = a + a + a + a or P = 4 * a.

3. Let the side square is equal to 4, that is, a = 3. Then the perimeter or length square, according to the resulting formula, will be equal to P = 4 * 3 or P = 12. The number 12 and will be the length or, which is the same, the perimeter square .

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Note!
The perimeter of a square is invariably correct, like any other length.

Useful advice
Similarly, it is allowed to detect the perimeter of a rhombus, because a square is a special case of a rhombus with right angles.

The perimeter characterizes the length of the closed silhouette. Like the area, it can be detected by other values ​​given in the problem statement. The tasks of finding the perimeter are extremely common in the school mathematics course.

Instructions

1. Knowing the perimeter and side of the figure, it is allowed to find its other side, as well as the area. The perimeter itself, in turn, can be detected along several specified sides or along the corners and sides, depending on the conditions of the problem. Also, in some cases, it is expressed through the area. The perimeter of the rectangle is especially primitive. Draw a rectangle with one side a and a diagonal d. Knowing these two quantities, find, according to the Pythagorean theorem, its other side, which is the width of the rectangle. Having found the width of the rectangle, calculate its perimeter in the following way: p = 2 (a + b). This formula is objective for all rectangles, since each of them has four sides.

2. Pay attention to the fact that the perimeter of a triangle in most problems is found if there is information about one of its angles. However, there are also problems in which all sides of the triangle are famous, and then the perimeter can be calculated by simple summation, without using trigonometric calculations: p = a + b + c, where a, b and c are sides. But such problems are rarely found in textbooks, because the method for solving them is clear. Solve more difficult tasks of finding the perimeter of a triangle in stages. Let's say draw an isosceles triangle, for which the base and angle are famous. In order to find its perimeter, first find sides a and b in a further way: b = c / 2cos ?. From the fact that a = b (an isosceles triangle), make a further total: a = b = c / 2cos ?.

3. Calculate the perimeter of a polygon in the same way, adding the lengths of all its sides: p = a + b + c + d + e + f and so on. If the polygon is positive and inscribed in or around a circle, calculate the length of one of its sides, and then multiply by their number. For example, in order to find the sides of a hexagon inscribed in a circle, proceed as follows: a = R, where a is the side of the hexagon equal to the radius of the circumscribed circle. Accordingly, if the hexagon is correct, then its perimeter is: p = 6a = 6R. If a circle is inscribed in a hexagon, then the side of the latter is: a = 2r? 3/3. Accordingly, find the perimeter of such a figure in a further way: p = 12r? 3/3.

Although the word "perimeter" comes from the Greek designation for a circle, it is customary to refer to it as the total length of the boundaries of any flat geometric figure, including a square. The calculation of this parameter, as usual, is not difficult and can be carried out by several methods, depending on the famous initial data.

Instructions

1. If we know the length of the side of a square (t), then to find its perimeter (p), primitively increase this value four times: p = 4 * t.

2. If the length of the side is unknown, but in the conditions of the problem the length of the diagonal (c) is given, then this is sufficient for calculating the length of the sides, and hence the perimeter (p) of the polygon. Use the Pythagorean theorem, which states that the square is the length of the long side right triangle(hypotenuse) is equal to the sum of the squares of the lengths of the short sides (legs). In a right-angled triangle made up of 2 adjacent sides of a square and connecting them extreme points segment, the hypotenuse coincides with the diagonal of the quadrilateral. It follows from this that the length of the side of the square is equal to the ratio of the length of the diagonal to the square root of two. Use this expression in the formula to calculate the perimeter from the previous step: p = 4 * c /? 2.

3. If only the area (S) of the perimeter of the square of the plane is given, then this will be enough in order to determine the length of one side. Because the area of ​​any rectangle is equal to the product of the lengths of its adjacent sides, then to find the perimeter (p), take the square root of the area, and quadruple the total: p = 4 *? S.

4. If the radius of the circle (R) described near the square is famous, then to find the perimeter of the polygon (p), multiply it by eight and divide the resulting total by the square root of two: p = 8 * R /? 2.

5. If the circle whose radius we know is inscribed in a square, then calculate its perimeter (p) by simply multiplying the radius (r) by an eight: P = 8 * r.

6. If the considered square in the conditions of the problem is described by the coordinates of its vertices, then to calculate the perimeter you only need data on 2 vertices belonging to one of the sides of the figure. Determine the length of this side, based on the same Pythagorean theorem for a triangle composed of itself and its projections on the coordinate axes, and quadruple the resulting total. Because the lengths of the projections onto the coordinate axes are equal to the modulus of the differences of the corresponding coordinates of 2 points (X?; Y? And X?; Y?), Then the formula can be written as follows: p = 4 *? ((X? -X?)? + (Y? -Y?)?).

In general, the perimeter is the length of the line that limits the closed figure. For polygons, the perimeter is the sum of all the side lengths. This value can be measured, and for many figures it is easy to calculate, if the lengths of the corresponding elements are known.

You will need

• - ruler or tape measure;
• - strong thread;
• - roller rangefinder.

Instructions

1. In order to measure the perimeter of an arbitrary polygon, measure all its sides with a ruler or other measuring device, and then find their sum. If you are given a quadrangle with sides of 5, 3, 7 and 4 cm, which are measured with a ruler, find the perimeter by adding them together P = 5 + 3 + 7 + 4 = 19 cm.

2. If the figure is arbitrary and includes not only straight lines, then measure its perimeter with a traditional rope or thread. To do this, place it so that it correctly repeats all the lines that bound the figure, and make a mark on it, if allowed, cut it off primitively in order to avoid confusion. After that, using a tape measure or ruler, measure the length of the thread, it will be equal to the perimeter of this figure. Be sure to make sure that the thread repeats the line as faithfully as possible for greater accuracy in the total.

3. Measure the perimeter of a difficult geometric figure with a roller rangefinder (curvimeter). To do this, a point is not marked on the line, at which the rangefinder roller is installed and rolled along it, until it returns to the starting point. The distance measured by the roller rangefinder will be equal to the perimeter of the figure.

4. Calculate the perimeter of some geometric shapes. Say, in order to find the perimeter of any positive polygon ( convex polygon whose sides are equal), multiply the length of the side by the number of corners or sides (they are equal). To discover the perimeter true triangle with a side of 4 cm, multiply this number by 3 (P = 4 × 3 = 12 cm).

5. In order to find the perimeter of an arbitrary triangle, add the lengths of all its sides. If not all sides are given, but there are angles between them, find them by the sine or cosine theorem. If two sides of a right-angled triangle are famous, find the third according to the Pythagorean theorem and find their sum. For example, if it is known that the legs of a right-angled triangle are 3 and 4 cm, then the hypotenuse will be equal to? (3? + 4?) = 5 cm.Then the perimeter P = 3 + 4 + 5 = 12 cm.

6. To find the perimeter of a circle, find the length of the circle that limits it. To do this, multiply its radius r by the number ?? 3.14 and the number 2 (P = L = 2 ??? r). If the diameter is known, consider that it is equal to two radii.

Perimeter polygon is called a closed polyline made up of all its sides. Finding the length of this parameter is reduced to summing the lengths of the sides. If all of the lines that make up the perimeter of such a two-dimensional geometric figure are identical in size, the polygon is said to be valid. In this case, the calculation of the perimeter is much easier.

Instructions

1. In the simplest case, when the length of the side (a) of the correct polygon and the number of vertices (n) in it, to calculate the length of the perimeter (P), primitively multiply these two values: P = a * n. Let's say the length of the perimeter of a true hexagon with a side of 15 cm should be 15 * 6 = 90 cm.

2. Calculate the perimeter of such polygon according to the known radius (R) of the circumscribed circle around it is also permissible. To do this, you will first have to express the length of the side using the radius and the number of vertices (n), and then multiply the resulting value by the number of sides. To calculate the length of the side, multiply the radius by the sine of Pi divided by the number of vertices, and double the total: R * sin (? / N) * 2. If you are more comfortable calculating the trigonometric function in degrees, replace Pi with 180 °: R * sin (180 ° / n) * 2. Calculate the perimeter by multiplying the resulting value by the number of vertices: P = R * sin (? / N) * 2 * n = R * sin (180 ° / n) * 2 * n. Say, if a hexagon is inscribed in a circle with a radius of 50 cm, its perimeter will have a length of 50 * sin (180 ° / 6) * 2 * 6 = 50 * 0.5 * 12 = 300 cm.

3. By a similar method, it is allowed to calculate the perimeter without knowing the length of the side of the positive polygon if it is circumscribed about a circle with a famous radius (r). In this case, the formula for calculating the size of the side of the figure will differ from the previous one only involved trigonometric function... Replace sine with tangent in the formula to get this expression: r * tg (? / N) * 2. Or for calculations in degrees: r * tg (180 ° / n) * 2. To calculate the perimeter, increase the value obtained by the number of times equal to the number peaks polygon: P = r * tg (? / N) * 2 * n = r * tan (180 ° / n) * 2 * n. Let's say the perimeter of an octagon, described near a circle with a radius of 40 cm, will be approximately equal to 40 * tg (180 ° / 8) * 2 * 8? 40 * 0.414 * 16 = 264.96 cm.

A square is a geometric figure consisting of four sides of identical length and four right angles, each of which is 90 °. Determining the area either perimeter quadrilateral, and any, is required not only when solving problems in geometry, but also in Everyday life... This knowledge can become useful, say, during repairs when calculating the required number of materials - flooring, walls or ceiling, as well as for laying out lawns and beds, etc.

Instructions

1. To determine the area of ​​a square, multiply the length by the width. Because the length and width are identical in a square, the value of one side is enough to square. Thus, the area of ​​a square is equal to the length of its side squared. The unit of measurement of the area can be square millimeters, centimeters, decimeters, meters, kilometers. To determine the area of ​​a square, it is allowed to use the formula S = aa, where S - area of ​​a square, and- side of a square.

2. Example No. 1. The room has the shape of a square. How much laminate (in square meters) is needed to completely cover the floor if the length of one side of the room is 5 meters. Write down the formula: S = aa. Substitute the data indicated in the condition in it. Because a = 5 m, therefore, the area will be equal to S (rooms) = 5x5 = 25 sq. M, which means that S (laminate) = 25 sq. M.

3. The perimeter is the total length of the shape's border. In a square, the perimeter is the length of all four, moreover, identical sides. That is, the perimeter of a square is the sum of all four of its sides. In order to calculate the perimeter of a square, it is enough to know the length of one of its sides. The perimeter is measured in millimeters, centimeters, decimeters, meters, kilometers. To determine the perimeter, there is the formula: P = a + a + a + a or P = 4a, where P is the perimeter and a is the length of the side.

4. Example No. 2. For finishing work in a square-shaped room, ceiling plinths are required. Calculate the total length (perimeter) of the skirting boards if the size of one side of the room is 6 meters. Write down the formula P = 4a. Substitute the data indicated in the condition into it: P (rooms) = 4 x 6 = 24 meters. Therefore, the length of the ceiling plinths will also be equal to 24 meters.

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Note!
For a square, the following definitions are objective: A square is a rectangle, one that has equal sides. A square is a special kind of a rhombus, in which all of the angles are 90 degrees. Being a positive quadrangle, it is allowed to describe or inscribe a circle around the square. The radius of a circle inscribed in a square can be found by the formula: R = t / 2, where t is the side of the square. If the circle is circumscribed around it, then its radius is found as follows: R = (? 2 * t) / 2 Based on these formulas, it is allowed output new ones to find the perimeter of the square: P = 8 * R, where R is the radius of the inscribed circle; P = 4 *? 2 * R, where R is the radius of the circumscribed circle. The square is a unique geometric figure, from the fact that it is unconditionally symmetrical, independently on how and where to draw the axis of symmetry.

Many remember what a square is from the school course. This quadrilateral, which is regular, has absolutely equal angles and sides. Looking around, you can see that we are surrounded by many squares. Every day we come across them, and sometimes it becomes necessary to find the area and perimeter of this geometric figure. Calculating these values ​​will not be difficult if you take a few minutes to watch this video tutorial explaining simple rules calculations.

Tutorial video “How to find the area and perimeter of a square”

What do you need to know about a square?

Before proceeding with the calculations, you need to know some important information about this figure, including:

• all sides of the square are equal;
• all corners of the square are straight;
• the area of ​​a square is a way of calculating how much space a figure takes up in two-dimensional space;
• 2D space is a piece of paper or a computer screen where a square is drawn;
• the perimeter is not an indicator of the fullness of the figure, but it allows you to work with its sides;
• the perimeter is the sum of all sides of the square;
• calculating the perimeter, we operate with one-dimensional space, which means fixing the result in meters, not square meters (area).

How do I find the area of ​​a square?

The calculation of the area of ​​a given figure can be simply and easily explained with an example:

• suppose the side of the square is 8 meters;
• to calculate the area of ​​any rectangle, you need to multiply the value of one side of it by the other (8 x 8 = 64);
• since we are multiplying meters by meters, the result is square meters (m2).

How do I find the perimeter of a square?

Knowing that all sides of a given rectangle are equal, you need to do the following manipulations to calculate its perimeter:

• add all four sides of the square (8 + 8 + 8 + 8 = 32);
• the resulting value will be the perimeter of the square, fixed in meters.

All formulas and calculus provided in this article are applicable to any rectangle. It is important to remember that when it comes to other rectangles that are not correct, the values ​​of the sides will be different, for example 4 and 8 meters. This means that in order to find the area of ​​such a rectangle, it will be necessary to multiply the sides of the figure that are different in value, and not the same.

It must also be remembered that the area is measured in square meters, and the perimeter is in simple meters. If the perimeter is drawn in the form of one long line, then its value will not change, which indicates that the calculations are carried out in one-dimensional space.

Area is measured in two-dimensional space, as indicated by square meters, which we get by multiplying meters by meters. Area is an indicator of the fullness of a geometric figure, and tells us how much imaginary coverage is needed to fill a square or other rectangle.

Simple explanations of the video lesson will allow you to quickly calculate the area and perimeter of not only a square, but also any rectangle. This knowledge of the school course will be useful when renovating a house or in a garden area.

Square - this is a geometric figure, which is a quadrangle, all angles and sides of which are equal. It can also be called rectangle whose adjacent sides are equal, or diamond where all angles are equal 90º... Thanks to the absolute symmetry to find square or perimeter of a square very easy.

Instructions:

• First, we define that perimeter called the sum of the lengths of all sides of a flat geometric figure, which is measured by the same quantities as the length. There are two ways to calculate the perimeter of a square.

Through side length and diagonal

• Because the perimeter of a square is determined by the sum of the lengths of all its sides, and the sides of this figure are equal, then you can calculate the value of this value by multiplying the length of one side by the number " 4 ". Accordingly, the formulas will look like this: P = a + a + a + a or P = a * 4 , where R- This perimeter of a square and but – side length.
• In addition, depending on the condition of the problem, the perimeter of a square can be calculated by multiplying the length of its diagonal by two roots of two: P = 2√2 * d , where R- This perimeter of a square and d- his diagonal.
• Some tasks require finding perimeter of a square knowing him square ... This is also not difficult to do. The area of ​​this figure is equal to the length of its side squared: S = a 2 , where S – square area and but –the length of its side... Or the area is equal to the square value of the length of its diagonal divided by two: S = d 2/2 , where S- still the same square and d – diagonal of a square.
• Knowing the formulas and the value of the area, it is not difficult to find the length of the side or the length of the diagonal, and then return to the formulas for calculating the perimeter and calculate its value.

Through the radius of the inscribed and circumscribed circle

• Finally, it is important to understand and how to find perimeter of a square if known circle radius described around it (or, on the contrary, inscribed in it). The circle inscribed in this geometric figure touches the middle of each side, and its radius is half of either side: R in = ½ a , where R in – inscribed circle radius and but – side of a square.
• Circumscribed circle passes through all the vertices of the square and its radius is half the length of the diagonal: R about = ½ d , where R oh - this radius of a circle circumscribed around a square and d- his diagonal.
• Therefore, in the first case, the perimeter will be calculated by the formula: P = 8 R in , and in the second: P = 4 x √2 x R about .

Using websites and an online calculator

• If for some reason you suddenly forgot the formulas, then the Internet will help you refresh your knowledge. Go to the browser, open the search engine page and type in the appropriate query in the window, for example: “ perimeter of a square formula". The system will give out a huge number sites of a reference nature that will help you in this matter, as well as allow you to cope with solving problems related to other geometric shapes.
• In addition, if you do not have a desire to understand the formulas and calculate the values ​​yourself, then you can use the services online calculators ... An example is the site. Chapter " Perimeter formulas for geometric shapes»Contains theoretical information supported by visual illustrations. If you follow the link “ online calculator ", Which is located in the window of each figure, then a page for calculations will open in front of you.
• Select in the window below, on the basis of which you are going to calculate perimeter of a square(side or diagonal) and then enter the available information. The system will issue result , guided by the established formulas.
• In addition, on the site you will find a lot of other information that can make it easier to work with mathematical problems... If you wish, you can search for more convenient or informative reference sites.
• If you cannot figure out the very course of solving the problem, then here you can turn to people who are good at solving mathematical exercises for help. They can always be found on the corresponding forums , for example, or.

This material contains geometric shapes with measurements. Measurements shown are approximate and may not match actual measurements. Lesson content

Perimeter of a geometric shape

The perimeter of a geometric figure is the sum of all its sides. To calculate the perimeter, you need to measure each side and add the measurements together.

Let's calculate the perimeter of the following figure:

This is a rectangle. We will talk about this figure in more detail later. Now let's just calculate the perimeter of this rectangle. Its length is 9 cm and its width is 4 cm.

A rectangle has equal opposite sides. This can be seen in the figure. If the length is 9 cm and the width is 4 cm, then the opposite sides will be 9 cm and 4 cm, respectively:

Let's find the perimeter. To do this, add all the sides. You can add them in any order, since the sum does not change from the rearrangement of the places of the terms. The perimeter is often indicated by a capital Latin letter P(eng. perimeters). Then we get:

P= 9 cm + 4 cm + 9 cm + 4 cm = 26 cm.

Since the opposite sides of the rectangle are equal, the finding of the perimeter is written down shorter - add the length and width, and multiply it by 2, which will mean "Repeat the length and width two times"

P= 2 × (9 + 4) = 18 + 8 = 26 cm.

A square is the same rectangle, but all sides are equal. For example, let's find the perimeter of a square with a side of 5 cm. Phrase "With the side 5cm" you need to understand how "The length of each side of the square is 5cm"

To calculate the perimeter, add all the sides:

P= 5 cm + 5 cm + 5 cm + 5 cm = 20 cm

But since all sides are equal, the calculation of the perimeter can be written as a product. The side of the square is 5 cm, and there are 4 such sides. Then this side, equal to 5 cm, must be repeated 4 times

P= 5 cm × 4 = 20 cm

Area of ​​a geometric figure

The area of ​​a geometric figure is a number that characterizes the size of a given figure.

It should be clarified that in this case we are talking about an area on a plane. A plane in geometry is called any flat surface, for example: a sheet of paper, a piece of land, a table surface.

Area is measured in square units Oh. By square units, we mean squares whose sides are equal to one. For example, 1 square centimeter, 1 square meter, or 1 square kilometer.

To measure the area of ​​a figure means to find out how many square units are contained in a given figure.

For example, the area of ​​the following rectangle is three square centimeters:

This is because this rectangle contains three squares, each of which has a side equal to one centimeter:

On the right is a square with a side of 1 cm (in this case, it is a square unit). If we look at how many times this square enters the rectangle on the left, we will find that it enters it three times.

The next rectangle has an area of ​​six square centimeters:

This is because this rectangle contains six squares, each of which has a side equal to one centimeter:

Let's say you wanted to measure the area of ​​the following room:

Let's decide in which squares we will measure the area. In this case, it is convenient to measure the area in square meters:

So, our task is to determine how many such squares with a side of 1 m are contained in the original room. Let's fill the whole room with this square:

We see that a square meter is contained in a room 12 times. This means that the area of ​​the room is 12 square meters.

Rectangle area

In the previous example, we calculated the area of ​​a room by sequentially checking how many times it contains a square whose side is equal to one meter. The area was 12 square meters.

The room was a rectangle. The area of ​​a rectangle can be calculated by multiplying its length and width.

To calculate the area of ​​a rectangle, you need to multiply its length and width.

Let's go back to the previous example. Let's say we measured the length of the room with a tape measure and it turned out that the length was 4 meters:

Now let's measure the width. Let it be 3 meters:

Multiply the length (4 m) by the width (3 m).

4 × 3 = 12

Like last time, we get twelve square meters. This is due to the fact that by measuring the length, we thereby find out how many times a square with a side equal to one meter can be laid in this length. Let's fit four squares into this length:

Then we determine how many times this length can be repeated with the stacked squares. We find out by measuring the width of the rectangle:

Square area

A square is the same rectangle, but all sides are equal. For example, the following figure shows a square with a side of 3 cm. Phrase "Square with side 3cm" means that all sides are equal to 3 cm

The area of ​​a square is calculated in the same way as the area of ​​a rectangle - the length is multiplied by the width.

We calculate the area of ​​a square with a side of 3 cm.Multiply the length of 3 cm by the width of 3 cm

In this case, it was required to find out how many squares with a side of 1 cm are contained in the original square. The original square contains nine squares with a side of 1 cm. Indeed, it is so. A square with a side of 1 cm enters the original square nine times:

Multiplying the length by the width, we got the expression 3 × 3, and this is the product of two identical factors, each of which is 3. In other words, the expression 3 × 3 is the second power of 3. So the process of calculating the area of ​​a square can be written as a power 3 2.

Therefore, the second power of a number is called square number... When calculating the second power of a number a, the person thereby finds the area of ​​a square with side a... The operation of raising a number to the second power is called differently squaring.

Designations

The area is denoted by a capital Latin letter S(eng. Square- square). Then the area of ​​a square with side a cm will be calculated according to the following rule

S = a 2

where a- the length of the side of the square. The second degree indicates that there is a multiplication of two identical factors, namely length and width. Earlier it was said that all sides of a square are equal, which means that the length and width of the square are equal, expressed through the letter a .

If the task is to determine how many squares with a side of 1 cm are contained in the original square, then cm 2 should be specified as the units of measurement for the area. This designation replaces the phrase "Square centimeter" .

For example, let's calculate the area of ​​a square with a side of 2 cm.

This means that a square with a side of 2 cm has an area equal to four square centimeters:

If the task is to determine how many squares with a side of 1 m are contained in the original square, then m 2 should be specified as the units of measurement. This designation replaces the phrase "square meter" .

We calculate the area of ​​a square with a side of 3 meters

This means that a square with a side of 3 m has an area equal to nine square meters:

Similar designations are used when calculating the area of ​​a rectangle. But the length and width of the rectangle can be different, so they are denoted with different letters, for example a and b... Then the area of ​​a rectangle with length a and width b is calculated according to the following rule:

S = a × b

As in the case of a square, the units of measure for the area of ​​a rectangle can be cm 2, m 2, km 2. These designations replace phrases "Square centimeter", "square meter", "square kilometer" respectively.

For example, let's calculate the area of ​​a rectangle 6 cm long and 3 cm wide

This means that a rectangle 6 cm long and 3 cm wide has an area equal to eighteen square centimeters:

It is allowed to use the phrase as a unit of measurement "Square units" ... For example, the entry S = 3 square unit means that the area of ​​a square or rectangle is equal to three squares, each of which has a unit side (1 cm, 1 m or 1 km).

Area unit conversion

Area units can be converted from one unit to another. Let's look at a few examples:

Example 1... Express 1 square meter in square centimeters.

1 square meter is a square with a side of 1 m. That is, all four sides have a length equal to one meter.

But 1 m = 100 cm. Then all four sides also have a length equal to 100 cm

Let's calculate the new area of ​​this square. Multiply the length 100 cm by the width 100 cm or square the number 100

S = 100 2 = 10,000 cm 2

It turns out that there are ten thousand square centimeters per square meter.

1 m 2 = 10,000 cm 2

This allows in the future to multiply any number of square meters by 10,000 and get the area expressed in square centimeters.

To convert square meters to square centimeters, you need to multiply the number of square meters by 10,000.

And in order to convert square centimeters to square meters, on the contrary, you need to divide the number of square centimeters by 10,000.

For example, let's translate 100,000 cm 2 into square meters. In this case, one can reason like this: “ if 10,000 cm 2 this is one square meter, then how many times 100,000 cm 2 will contain 10,000 cm 2 "

100,000 cm 2: 10,000 cm 2 = 10 m 2

Other units of measurement can be converted in the same way. For example, let's translate 2 km 2 into square meters.

One square kilometer is a square with a side of 1 km. That is, all four sides are one kilometer long. But 1 km = 1000 m. This means that all four sides of the square are also 1000 m. Let's find the new area of ​​the square, expressed in square meters. To do this, multiply the length of 1000 m by the width of 1000 m or square the number 1000

S = 1000 2 = 1,000,000 m 2

It turns out that there is one million square meters per square kilometer:

1 km 2 = 1,000,000 m 2

This allows in the future to multiply any number of square kilometers by 1,000,000 and get the area expressed in square meters.

To convert square kilometers to square meters, you need to multiply the number of square kilometers by 1,000,000.

So, back to our task. It was required to translate 2 km 2 into square meters. Multiply 2 km 2 by 1,000,000

2 km 2 × 1,000,000 = 2,000,000 m 2

And to convert square meters to square kilometers, on the contrary, you need to divide the number of square meters by 1,000,000.

For example, let's translate 3,500,000 m 2 into square kilometers. In this case, one can reason like this: “ if 1,000,000 m 2 is one square kilometer, then how many times 3,500,000 m 2 will contain 1,000,000 m 2 "

3,500,000 m 2: 1,000,000 m 2 = 3.5 km 2

Example 2... Express 7 m2 in square centimeters.

Multiply 7 m2 by 10,000

7 m 2 = 7 m 2 × 10,000 = 70,000 cm 2

Example 3... Express 5 m 2 13 cm 2 in square centimeters.

5 m 2 13 cm 2 = 5 m 2 × 10,000 + 13 cm 2 = 50,013 cm 2

Example 4... Express 550,000 cm 2 in square meters.

Let's find out how many times 550,000 cm 2 contains 10,000 cm 2. To do this, divide 550,000 cm 2 by 10,000 cm 2

550,000 cm 2: 10,000 cm 2 = 55 m 2

Example 5... Express 7 km 2 in square meters.

Multiply 7 km 2 by 1,000,000

7 km 2 × 1,000,000 = 7,000,000 m 2

Example 6... Express 8,500,000 m 2 in square kilometers.

Let's find out how many times 8,500,000 m 2 contains 1,000,000 m 2 each. To do this, we divide 8,500,000 m 2 by 1,000,000 m 2

8,500,000 m 2 × 1,000,000 m 2 = 8.5 km 2

Units of measurement for the area of ​​land plots

It is convenient to measure the area of ​​small land plots in square meters.

Larger land plots are measured in macaws and hectares.

Ar(abbreviated: a) Is an area equal to one hundred square meters (100 m 2). In view of the frequent spread of such an area (100 m 2), it began to be used as a separate unit of measurement.

For example, if it is said that the area of ​​some field is 3 a, then you need to understand that these are three squares with an area of ​​100 m 2 each, that is:

3 a = 100 m 2 × 3 = 300 m 2

Among the people ar often call weaving since ap equal to square, with an area of ​​100 m 2. Examples:

1 weaving = 100 m 2

2 ares = 200 m 2

10 ares = 1000 m 2

Hectare(abbreviated: ha) is an area equal to 10,000 m 2. For example, if it is said that the area of ​​some forest is 20 hectares, then you need to understand that this is twenty squares with an area of ​​10,000 m 2 each, that is:

20 ha = 10,000 m 2 × 20 = 200,000 m 2

Rectangular parallelepiped and cube

A rectangular parallelepiped is a geometric shape made up of faces, edges, and vertices. The figure shows a rectangular parallelepiped:

Shown in yellow facets parallelepiped, in black - ribs, red - tops.

A rectangular parallelepiped has a length, width, and height. The figure shows where the length, width and height are:

A parallelepiped whose length, width and height are equal to each other is called. The figure shows a cube:

The volume of a geometric figure

The volume of a geometric figure Is a number that characterizes the capacity of this figure.

Volume is measured in cubic units. Cubic units mean cubes 1 long, 1 wide and 1 high. For example, 1 cubic centimeter or 1 cubic meter.

To measure the volume of a figure means to find out how many cubic units fit in a given figure.

For example, the volume of the following rectangular parallelepiped is equal to twelve cubic centimeters:

This is because this parallelepiped holds twelve cubes 1 cm long, 1 cm wide and 1 cm high:

The volume is indicated by a capital Latin letter V... One of the units of measure for volume is the cubic centimeter (cm 3). Then the volume V the parallelepiped considered by us is 12 cm 3

V= 12 cm 3

The volume of any parallelepiped is calculated as follows: multiply its length, width and height.

The volume of a rectangular parallelepiped is equal to the product of its length, width and height.

V = abc

where, a- length, b- width, c- height

So, in the previous example, we visually determined that the volume of the parallelepiped is 12 cm 3. But you can measure the length, width and height of a given parallelepiped and multiply the measurement results. We will get the same result

Volume is calculated in the same way as volume rectangular parallelepiped- multiply the length, width and height.

For example, let's calculate the volume of a cube whose length is 3 cm. The length, width and height of a cube are equal to each other. If the length is 3 cm, then the width and height of the cube are equal to the same three centimeters:

We multiply the length, width, height and we get a volume equal to twenty-seven cubic centimeters:

V= 3 × 3 × 3 = 27 cm³

Indeed, the original cube contains 27 cubes 1 cm long

When calculating the volume of this cube, we multiplied the length, width and height. The product is 3 × 3 × 3. This is the product of three factors, each of which is 3. In other words, the product 3 × 3 × 3 is the third power of 3 and can be written as 3 3.

V= 3 3 = 27 cm 3

Therefore, the third power of a number is called cube numbers... When calculating the third power of a number a, a person thereby finds the volume of a cube, length a... The operation of raising a number to the third power is called differently cube.

Thus, the volume of a cube is calculated according to the following rule:

V = a 3

Where a - the length of the cube.

Cubic decimeter. Cubic meter

Not all objects in our world are conveniently measured in cubic centimeters. For example, it is more convenient to measure the volume of a room or house in cubic meters (m 3). And the volume of a tank, aquarium or refrigerator is more convenient to measure in cubic decimeters (dm 3).

Another name for one cubic decimeter is one liter.

1 dm 3 = 1 liter

Volume unit conversion

Volume units can be converted from one unit to another. Let's look at a few examples:

Example 1... Express 1 cubic meter in cubic centimeters.

One cubic meter is a cube with a side of 1 m. The length, width and height of this cube are equal to one meter.

But 1 m = 100 cm. This means that the length, width and height are also 100 cm.

Let's calculate the new volume of the cube, expressed in cubic centimeters. To do this, multiply its length, width and height. Or we will cube the number 100:

V = 100 3 = 1,000,000 cm 3

It turns out that there is one million cubic centimeters per cubic meter:

1 m 3 = 1,000,000 cm 3

This allows in the future to multiply any number of cubic meters by 1,000,000 and get the volume expressed in cubic centimeters.

To translate Cubic Meters in cubic centimeters, you need to multiply the number of cubic meters by 1,000,000.

And to convert cubic centimeters to cubic meters, on the contrary, you need to divide the number of cubic centimeters by 1,000,000.

For example, let's translate 300,000,000 cm 3 into cubic meters. In this case, one can reason like this: “ if 1,000,000 cm 3 this is one cubic meter, then how many times 300,000,000 cm 3 will contain 1,000,000 cm 3 "

300,000,000 cm 3: 1,000,000 cm 3 = 300 m 3

Example 2... Express 3 m 3 in cubic centimeters.

Multiply 3 m 3 by 1,000,000

3 m 3 × 1,000,000 = 3,000,000 cm 3

Example 3... Express 60,000,000 cm 3 in cubic meters.

We find out how many times 60,000,000 cm 3 contains 1,000,000 cm 3. To do this, divide 60,000,000 cm 3 by 1,000,000 cm 3

60,000,000 cm 3: 1,000,000 cm 3 = 60 m 3

The capacity of a tank, can or canister is measured in liters. Liter is also a unit of measure for volume. One liter is equal to one cubic decimeter.

1 liter = 1 dm 3

For example, if the capacity of a can is 1 liter, this means that the volume of this can is 1 dm 3. When solving some problems, it may be useful to be able to convert liters to cubic decimeters and vice versa. Let's look at a few examples.

Example 1... Convert 5 liters to cubic decimeters.

To convert 5 liters to cubic decimeters, just multiply 5 by 1

5 l × 1 = 5 dm 3

Example 2... Convert 6000 liters to cubic meters.

Six thousand liters is six thousand cubic decimeters:

6000 l × 1 = 6000 dm 3

Now let's translate these 6000 dm 3 into cubic meters.

The length, width and height of one cubic meter are equal to 10 dm

If we calculate the volume of this cube in decimeters, we get 1000 dm 3

V= 10 3 = 1000 dm 3

It turns out that one thousand cubic decimeters corresponds to one cubic meter. And in order to determine how many cubic meters correspond to six thousand cubic decimeters, you need to find out how many times 6,000 dm 3 contains 1,000 dm 3

6,000 dm 3: 1,000 dm 3 = 6 m 3

This means that 6000 l = 6 m 3.

Square table

In life, you often have to find the areas of various squares. To do this, each time you need to raise the original number to the second power.

First 99 squares natural numbers have already been calculated and entered into a special table called table of squares.

The first row of this table (numbers from 0 to 9) is the original number, and the first column (numbers from 1 to 9) is the original number.

For example, let's find the square of the number 24 from this table. The number 24 consists of the numbers 2 and 4. More precisely, the number 24 consists of two tens and four ones.

So, select number 2 in the first column of the table (column of tens), and select number 4 in the first row (row of units). Then, moving to the right of the number 2 and down from the number 4, we find the intersection point. As a result, we will find ourselves in the position where the number 576 is located. This means that the square of the number 24 is the number 576

24 2 = 576

Cubes table

As in the situation with squares, the cubes of the first 99 natural numbers have already been calculated and entered into a table called cubes table.

Calculate the volume of a rectangular parallelepiped 6 cm long, 4 cm wide, 3 cm high. Problem 7. Areas land plot, sown with wheat and flax, are proportional to the numbers 4 and 5. In what area is wheat sown, if 15 hectares are sown under flax

Solution

The number 4 represents the area planted with wheat. And the number 5 reflects the area sown with flax.
The area planted with wheat and flax is said to be proportional to these numbers.

Simply put, how many times the numbers 4 or 5 change, how many times will the area sown with wheat or flax also change. Flax is sown on 15 hectares. That is, the number 5, which reflects the area sown with flax, has changed 3 times.

Then the number 4, which reflects the area sown with wheat, needs to be tripled.

4 × 3 = 12 ha

Answer: wheat is sown on 12 hectares.

Problem 8. The length of the granary is 42 m, the width is length, and the height is 0.1 of the length. Determine how many tons of grain a granary holds if 1 m 3 of it weighs 740 kg.

Solution

Let's determine how many liters per minute are poured through the second pipe:

25 l / min × 0.75 = 18.75 l / min

Let's determine how many liters per minute are poured into the pool through both pipes:

25 l / min + 18.75 l / min = 43.75 l / min

Determine how many liters of water will be poured into the pool in 13 hours 32 minutes

43.75 x 13 h 32 min = 43.75 x 812 min = 35,525 liters

1 l = 1 dm 3

35 525 l = 35 525 dm 3

Let's convert cubic decimeters to cubic meters. This will calculate the volume of the pool:

35 525 dm 3: 1000 dm 3 = 35.525 m 3

Knowing the volume of the pool, you can calculate the height of the pool. Substitute in the literal equation V = abc the meanings we have. Then we get:

V = 35,525
a = 5.8
b = 3.5
c= x

35.525 = 5.8 × 3.5 × x
35.525 = 20.3 × x
x= 1.75 m

c = 1.75

Answer: the height (depth) of the pool is 1.75 m.

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