How to make the right pentagon. How to build a pentagon with a circulation

Dictionary Ozhegova says that the pentagon is a limited five intersecting straight, forming five internal angles, as well as any item of such a form. If this polygon has the same sides and angles, then it is called the right (Pentagon).

Than interesting right pentagon?

It was in such a form that the well-known building of the United States Ministry of Defense was built. From volumetric right polyhedra Only a dodecahedron has a face in the form of a pentagon. And in nature, there are no crystals, the faces of which resembled the right pentagon. In addition, this figure is a polygon with a minimum number of corners, which cannot be coated area. Only a pentagon number of diagonals coincides with the number of its parties. Agree, it's interesting!

Basic properties and formulas

Using formulas for an arbitrary right polygon, you can define all the necessary parameters that the Pentagon has.

• The central angle α \u003d 360 / n \u003d 360/5 \u003d 72 °.
• The inner angle β \u003d 180 ° * (N - 2) / n \u003d 180 ° * 3/5 \u003d 108 °. Accordingly, the sum of the internal angles is 540 °.
• The diagonal ratio towards the side is equal to (1 + √5) / 2, that is, (approximately 1,618).
• The length of the parties that the right pentagon has, can be calculated by one of three formulas, depending on which parameter is already known:

• if the circle is described around it and its radius R is known, then A \u003d 2 * R * Sin (α / 2) \u003d 2 * R * Sin (72 ° 2) ≈1,1756 * R;
• in the case when the circle with radius R is inscribed in the correct pentagon, A \u003d 2 * R * TG (α / 2) \u003d 2 * R * TG (α / 2) ≈ 1.453 * R;
• it happens that instead of radii, the value of the diagonal D is known, then the side is determined as follows: a ≈ D / 1.618.

• The area of \u200b\u200bthe correct pentagon is determined, again, depending on which parameter is known to us:
• if there is an inscribed or described circle, then one of two formulas is used:

S \u003d (n * a * r) / 2 \u003d 2.5 * a * r or s \u003d (n * r 2 * sin α) / 2 ≈ 2.3776 * R 2;

• the area can also be determined, knowing only the length of the side A:

S \u003d (5 * A 2 * TG54 °) / 4 ≈ 1.7205 * A 2.

Proper Pentagon: Building

This geometric shape You can build differently. For example, enter it into a circle with a given radius or to build on the basis of a given side. The sequence of actions was described in the "beginning of" Euclidea of \u200b\u200babout 300 years BC. In any case, we will need a circulation and a ruler. Consider a method of constructing with a given circumference.

1. Choose an arbitrary radius and draw a circle, indicating its center point O.

2. On the circle line, select a point that will serve as one of the tops of our pentagon. Let it be point A. Connect points about and but a direct segment.

3. Swipe straight through the point about perpendicular to the direct OA. The location of the intersection of this straight line with the line of the circle mark as point B.

4. In the middle of the distance between points O and in Build the point S.

5. Now draw a circle, the center of which will be at the point C and which will pass through the point A. The place of its intersection with a straight OB (it will be inside the very first circle) will be dot D.

6. Build a circle passing through D, the center of which will be in A. Places of its intersection with the initial circle must be denoted by points E and F.

7. Now build a circle, the center of which will be in E. Make it necessary so that it goes through A. It is another place for intersection of the original circle need to be denoted

8. Finally, build a circle through a with a center at point F. Mark another place of intersection of the original circumference with the point H.

9. Now it remains only to connect the vertices A, E, G, H, F. Our correct pentagon will be ready!

Difficulty level: easy

1 step

First, choose where to place the center of the circle. It is necessary to put the starting point, let it be called O. with the help of a circulation, draw a circle of a given diameter or radius around it.

2 step

Then we spend two axes through the point O, the center of the circle, one horizontal, the other is 90 degrees with respect to it - vertical. The intersection points horizontally call on the left to the right A and B, vertically, from top to bottom - M and N. Radius, which lies on any axis, for example, on the horizontal in the right part, divide in half. This can be done like this: a circulation with a radius of the circumference known to us, we set the edge to the point of crossing the horizontal axis and the circle - in, we draw the intersection with the circle, call the points, respectively, from top to bottom - C and P, connect them to the segment that will cross the axis axis, I call the intersection point K.

3 Step

We connect the points to and m and we obtain a segment of the CM, we set the circulation to the point M, we set the distance to the point to the point to and outline the labels on the OA radius, we call this point e, then we are measured to cross the circle to the intersection with the left upper part of the OM circle. This point of intersection is called F. The distance equal to the segment of me is the desired side of the equilateral pentagon. At the same time, the point M will be one vertex of the embedded pentagon circumference, and the point F is another.

4 Step

Further, from the points obtained along the entire circumference, we draw a circle of distances equal to the segment of me, all points should turn out to be 5. We connect all the points in segments - we get a pentagon, inscribed in a circle.

• When drawing, be careful in distances in measurements, do not allow the errors so that the pentagon really hurts the equilateral

First method - on this side s using the transport.

We carry out the straight and postpone on it AB \u003d S; We accept this line for the radius and this radius from points A and in describe arcs: Next, using the TRANSPORT, we build at these points of the angles at 108 °, the sides of which will cross with arcs at points C and D; From these points with a radius of AB \u003d 5, describe arcs that cross in e, and connect the lines of the lines, C, E, D, V.

The resulting pentagon - the desired one.

The second way. We carry out a circle with radius r. From the point and the circulation, we carry out an arc of the radius AM to the intersection at points in and from the circle. Connect in and with a line that cross the horizontal axis at the point E.

Then we carry out an arc from point E, which will cross the horizontal line at the point O. Describe, finally, from the point F arc, which will cross the circle at the points N and K. Putting the distance FO \u003d FH \u003d Fk in the circumference and connecting the division points with lines We get the right pentagon.

Third way. In this circle, enter the right pentagon. We carry out two mutually perpendicular diameters of AB and MS. We divide the radius of JSC Point E Poland. From the point E, as from the center, we carry out an arc of the circle of the radius and we interfere with the diameter of AV at the point F. The MF segment is equal to the side of the desired proper pentagon. Circular solution equal to MF, we make a serif N 1, P 1, Q 1, to 1 and connect them straight.

The figure built a hexagon on this side.

Straight AB \u003d 5, as a radius, from points A and in describe arcs that cross in C; From this point, the same radius describes the circle on which the party and will be postponed 6 times.

Hexagon Adefgb.- the desired one.

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The task of constructing a faithful pentagon is reduced to the task of dividing the circle to five equal parts. From the fact that the faithful pentagon is one of the figures containing the proportions of the golden section, paints and mathematicians were interested in its construction. Now several methods for constructing a faithful polygon, inscribed in a given circle, are discovered.

You will need

• - ruler
• - Circul

Instruction

1. Apparently, if we build a faithful decada, and then to combine the peaks through one, then we get a pentagon. To build a decidagon, draw the circumference of the specified radius. Indicate its center of the letter O. Swipe two perpendicular radius, in the figure, they are indicated as OA1 and OB. OB Radius Wheel with a subfolder or a method of dividing a segment of pressure with a force of a circulation. Build a small circle with a center C in the middle of the segment OB by a radius equal to half OB. Tighten the point C with a point A1 on the initial circumference according to a ruler. The CA1 segment intersects the auxiliary circle at D. DA1 segment is equal to the side of the faithful decada, inscribed in this circle. Circle to notice this segment on the circle, then combine the intersection points through one and you will get a positive pentagon.

2. Another method discovered the German artist Albrecht Durer. In order to build a pentagon on its method, start again with the construction of the circle. Reconnect its center O and spend two perpendicular radius of OA and OB. OA Radius Welcome in the middle of the letter C. Install the circulant needle to the point C and open it to point B. Spend the circle of the BC radius to the intersection with the diameter of the initial circle on which the OA radius lies. The intersection point indicates D. Cut BD - side of a positive pentagon. Set this segment five times on the initial circle and combine the intersection points.

3. If you need to build a pentagon on its specified side, then you needed the 3rd method. In the line of the pentagon side, mark this segment with letters a and B. divide it on 6 equal parts. From the middle of the cut AB, swipe, perpendicular segment. Build two circumference with AB radius and centers in A and B, as if you were going to divide the segment of pressure. These circumference intersect at the point C. Point C is on the ray, which is proceeded perpendicularly up from the middle of the AB. Put the distance equal to 4/6 from the length of this ray, mark this point D. Build the AB radius circle with the center at point D. The intersection of this circle with two auxiliary built previously will give the last two vertices of the pentagon.

The topic of dividing the circumference to equal parts in order to build the faithful inscribed polygons has long been occupied by the minds of ancient scientists. These theses of constructing using a circulation and ruler were expressed even in the Euclidean "principles". But only two millennia, this task was completely solved not only graphically, but also mathematically.

Instruction

1. Approximate construction positive pentagon By the method of A. Dürera, with a force of a circulation and a ruler (through two circles with a universal radius equal to the side pentagon).

2. Building loyal pentagon Based on a positive decagound, inscribed in a circle (combining the tops of a decidagon through one).

3. Graphic construction through the calculated inner corner pentagon With the support of the transport and the ruler (the sum of the angles of the convex N-carbon is Sn \u003d 180 ° (n - 2), because at a positive polygon, all the corners are equal). For n \u003d 5, S5 \u003d 5400, then the value of the angle of 1080.a is also with the support of the circle and 2 beams emerging from its center, provided that the angle between them is 720, since (36005 \u003d 720). Their intersection with a circle will give a segment equal to the side pentagon .

4. Another light graphic method: to divide the diameter of the preset AB circle into three parts (AC \u003d CD \u003d DE). From the point D, omit perpendicular to the intersection with a circle at points E, F.Provypiy through the sections of EC and FC before crossing with a circle, we obtain a point G, H. G, E, B, F, H - the tops of the positive pentagon .

5. Building with support for reception of bion (allowing the faithful in the circumference of the polygon inscribed into the circumference with any number of sides N at a given ratio). Let's find: for n \u003d 5. We erected a positive triangle ABC, where AB is the diameter of the specified circle. Detect on the AB point D, for further relation: AD: AB \u003d 2: N. At n \u003d 5, ad \u003d 25 * AB. We will spend directly through the CD to intersection with a circle at point E. Segment AE - the side of the right thing inscribed pentagon . Print n \u003d 5,7,9,10 Build error does not exceed 1%. With increasing N, the error of approximation is growing, but it remains smaller 10.3%.

6. Building according to a given side according to the method of L. da Vinci (applying the relation between the side of the polygon (AN) and Apophistician (HA): AN / 2: HA \u003d 3 / (n-1), which is allowed to express as follows: TG180 ° / N \u003d 3 / (n-1)).

7. The general method of constructing positive polygons according to the specified side by the F. Kowarzhik method (1888), based on the rules of L. Ya Vinci. The pre-constructing of a positive N-carbon on the basis of the Falez theorem. Advanced add only that approximate methods for building polygons are genuine, Primitive and beautiful.

There are two basic methods of building a faithful polygon with five sides. Both of them believe the use of a circulation, line and pencil. 1st method is the fitting pentagon In a circle, and the 2nd method is based on a given length of the side of your coming geometric shape.

You will need

• Circle, line, pencil

Instruction

1. 1st construction method pentagon It is considered more "typical." To begin, build the circle and somehow designate its center (usually the letter O) is used for this). After that, spend the diameter of this circle (let's call it) and divide one of the 2-received radii (say, OA) exactly input. The middle of this radius is denoted by the letter S.

2. From the point O (center of the initial circle), spend another radius (OD), the one that will be severely perpendicled by the previously conducted diameter (AV). After that, take the circus, put it into the point C and measure the distance to the intersection of a new radius with a circle (CD). The same distance to postpone on the diameter of AB. You will get a new point (let's call it e). Measure the distance from the point D to the point E - it will be equal to the length of the side of your coming pentagon .

3. Put the circuit to the point D and set the distance on the circle, equal to the segment of DE. Repeat this procedure for another 3 times, and then combine the point D and 4 new points on the initial circle. The figure in the result of the construction will be a faithful pentagon.

4. In order to build a pentagon with a different method, to start, draw a segment. Let's say it will be a length of 9 cm. Next, divide your segment to 6 equal parts. In our case, the length of any part will be 1.5 cm. Now take the circus, put it in one of the ends of the segment and take a circle or an arc with a radius equal to the length of the segment (AV). After that, rearrange the circus to another end and repeat the operation. The resulting circles (or arcs) will cross at one point. Let's call it C.

5. Now take a ruler and spend directly through the point with and the center of the ab. After that, starting from the point with set aside on this straight line, constituting 4/6 of the AB segment. The 2nd end of the segment is denoted by the letter D. Point D will be one of the vertices of the coming pentagon . From this point, swipe the circle or arc with a radius equal to av. This circle (arc) will cross the circles (arcs) previously built by you, which are two missing vertices. pentagon . Combine these points with the vertices D, A and B, and building a positive pentagon will be completed.

Video on the topic

Ray - This is a straight line conducted from the point and does not have the end. There are other definitions of the beam: let's say, "... This is a straight line, limited to the point on the one hand." How to positively draw a beam and what accessories to do you need?

You will need

• Sheet of paper, pencil and ruler.

Instruction

1. Take a sheet of paper and check in an arbitrary location. After that, attach the line and swipe the line starting from the specified point and to infinity. This drawn line is referred to as the beam. Now let's notice another point on the beam, for example, the letter C. The line from the original and to the point C will be called the segment. If you primitively draw the line and not notice the truth to one point, then this direct will not be a beam.

2. Draw a beam in any graphic editor or in the same MSOffice is no more difficult than manually. For example, take the Microsoft Office 2010 program. Go to the "Insert" section and select the "Figures" element. In the drop-down list, select the "line" figure. Further the cursor takes the type of cross. In order to draw a flat line, press the "SHIFT" key and swipe the required length. Immediately later, the format tab opens. Now you have a primitive straight line and there is no fixed point, and based on the definition, the beam must be limited to the point on one side.

3. In order to make the point at the beginning of the line, do the following: Highlight the drawn line and call the context menu by pressing the right mouse button.

4. Select "Figure Format". On the left menu, select "Line Type". Further detect the "Line Settings" header and select "Type of start" as a circle. There you can set up the thickness of the start and end lines.

5. Remove the selection from the line and see that the point appeared at the beginning of the line. To create an inscription, click the "Draw an inscription" button and make the field where the inscription will be. Later writing an inscription click on free place And it is activated.

6. The beam has a safely painted and took it all a few minutes. The drawing of the beam in other editors is carried out by the same thesis. When the "SHIFT" key is pressed, proportional figures will invariably. Nice use.

Video on the topic

Note!
The ratio of the diagonal of the faithful pentagon to its side is golden cross section (irrational number (1 + √5) / 2). And the five inner angles of the pentagon are 108 °.

Helpful advice
If you combine the vertices of the faithful pentagon diagonals, then the pentagram will be.

Building inscribed in the circumference of the correct hexagon.

The construction of the hexagon is based on the fact that its side is equal to the radius of the circle described. Therefore, for the construction, it is enough to divide the circle to six equal parts and connect the found points to each other.

The right hexagon can be built using a 30x60 ° coal. To perform this construct, we accept the horizontal diameter of the circle for bisector of the angles 1 and 4, build the sides 1 - 6, 4 - 3, 4 - 5 and 7 - 2, after which we carry out the parties 5 - 6 and 3 - 2.

The vertices of such a triangle can be constructed using a circulation and angle with angles of 30 and 60 ° or only one circular. Consider two ways to build inscribed in the circumference of the equilateral triangle.

First method (FIG. 61, A) is based on the fact that all three angle of the triangle 7, 2, 3 contains 60 °, and the vertical straight line, conducted through point 7, is simultaneously height and bisector angle 1. Since angle 0 - 1 - 2 is 30 °, then to find the side 1 - 2, it is sufficient to build at point 1 and side 0 - 1 an angle of 30 °. To do this, we establish a flight and a square as shown in the figure, carry out a line 1 - 2, which will be one of the sides of the desired triangle. To construct the side 2 - 3, we set the flight to the position shown by the dash lines, and through the point 2 we carry out a straight line that will determine the third vertex of the triangle.

Second way It is based on the fact that, if you build a regular hexagon, inscribed in a circle, and then combine its vertices through one, then the equilateral triangle will be.

To build a triangle, we plan on the diameter of the vertex point 1 and carry out the diametrical line 1 - 4. Next from point 4 by a radius equal to D / 2, describe an arc before intersection with a circle at points 3 and 2. The points obtained will be two other vertices of the desired triangle.

This construction can be performed using a square and a circular.

First method It is based on the fact that the diagonal of the square intersect in the center of the described circle and tilted to its axes at an angle of 45 °. Based on this, we establish a reynchin and angle with angle of 45 ° as shown in FIG. 62, and, and we note the points 1 and 3. Next through these points, we carry out the horizontal sides of the square 4 - 1 and 3 -2 using the reiser. Then, with the help of a coal cathet, we spend the vertical sides of the square 1 - 2 and 4 - 3.

Second way It is based on the fact that the vertices of the square are divided by in half an arc circumference enclosed between the ends of the diameter. We plan at the ends of two mutually perpendicular diameters of the point A, B and C and from them radius, we describe the arc to the mutual intersection of them.

Next, through the points of intersection of arcs, we carry out auxiliary straight lines marked on the figure with solid lines. The points of their intersection with the circle will determine the vertices 1 and 3; 4 and 2. Thus, the vertices of the desired square are combined consistently between them.

Building inscribed in the circumference of the right pentagon.

To enter the correct pentagon in the circumference, we produce the following constructions. We look at the circumference point 1 and accept it for one of the tops of the pentagon. We divide the segment of JSC in half. For this, the radius of AO from the point A is described by the arc before intersection with the circle at the points M and B. By connecting these points directly, we obtain the point K, which can then connect with a point 1. Radius equal to the segment A7, describe from a point to an arc before intersection from diametral AO line at the point H. By connecting point 1 with a point H, we get the side of the pentagon. Then, with a solution of a circulation, equal to a segment of 1H, describing the arc from the vertex 1 to the intersection with a circle, we find the vertices 2 and 5. When making a sort of vertex 2 and 5, we obtain the remaining vertices 3 and 4. The points found sequentially connect each other.

Building the right pentagon on this side.

To build a right pentagon on this side (FIG. 64), we divide the AB segment to six equal parts. From the points A and in the AB radius, we describe the arc, the intersection of which will give the point K. through this point and division 3 on a straight AB carry out a vertical direct. Next, from the point to on this straight, lay the segment equal to 4/6 AB. We will get the point 1 -Thero pentagon. Then, with a radius equal to AB, from point 1, describe an arc before intersection with arcs previously carried out from the points A and B. The point of intersection of the arcs determine the vertices of the pentagon 2 and 5. The vertices found connect each other.

Building inscribed in the circumference of the right sevenginous.

Let the circumference of the diameter D; It is necessary to enter in it the right sevenfone (Fig. 65). We divide the vertical diameter of the circle to seven equal parts. From a point 7 by a radius equal to the diameter of the circumference D, describe the arc to the intersection with the continuation of the horizontal diameter at the point F. The point f will call the pole of the polygon. Taking a point VII for one of the vertices of the semi-angry, we carry out from the pole of F through a clear division of the vertical diameter of the rays, the intersection of which with a circle will determine the vertices VI, V and IV of the sevendent. To obtain vertices / - // - /// From points IV, V and VI, we spend the horizontal straight lines to cross the circle. The vertices found are connecting between themselves. The sevenfoon can be built by conducting rays from the pole f and through the odd divisions of the vertical diameter.

The given method is suitable for the construction of the right polygons with any number of parties.

The division of the circle on any number of equal parts can also be made using the data of the table. 2, in which the coefficients give the ability to determine the size of the sides of the right inscribed polygons.

The length of the sides of the right inscribed polygons.

In the first column of this table, the numbers of the sides of the correct inscribed polygon are indicated, and in the second - coefficients. The length of the sides of a given polygon will succeed on multiplying the radius of this circle to the coefficient corresponding to the number of the sides of this polygon.