Correct pentagon: the minimum information required

Explanatory dictionary Ozhegova says that the pentagon is a geometric figure, bounded by five intersecting lines, forming five internal corners, as well as any object of a similar shape. If a given polygon has all the sides and angles identical, then it is called the correct (pentagon).

What is the interest of a regular pentagon?

It was in this form that the famous building of the Ministry of Defense of the United States was built. Of the volume regular polyhedra, only a dodecahedron has faces in the form of a pentagon. And in nature there are no crystals at all, the faces of which would resemble a regular pentagon. In addition, this figure is a polygon with a minimum number of angles, which it is impossible to square the area. Only at the pentagon the number of diagonals coincides with the number of its sides. Agree, it's interesting!

Basic Properties and Formulas

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

• The central angle is α = 360 / n = 360/5 = 72 °.
• The internal angle β = 180 ° * (n-2) / n = 180 ° * 3/5 = 108 °. Correspondingly, the sum of the internal angles is 540 °.
• The ratio of the diagonal to the side is (1 + √5) / 2, that is, the "golden section" (about 1.618).
• The length of the side that the regular pentagon has can be calculated according to one of the three formulas, depending on which parameter is already known:

• If a circle is circumscribed around it and its radius R is known, then a = 2 * R * sin (α / 2) = 2 * R * sin (72 ° / 2) ≈ 1.1756 * R;
• In the case when a circle with radius r is inscribed in a regular pentagon, a = 2 * r * tg (α / 2) = 2 * r * tg (α / 2) ≈ 1.453 * r;
• It happens that instead of radii the diagonal value D is known, then the side is determined as follows: a ≈ D / 1,618.

• The area of the regular pentagon is determined, again, depending on which parameter is known to us:
• If there is an inscribed or circumscribed circle, then one of two formulas is used:

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

• The area can also be determined knowing only the length of the lateral side a:

S = (5 * a ² * tg54 °) / 4 ≈ 1.7205 * a ² .

Correct pentagon: construction

This geometric figure can be constructed in different ways. For example, write it in a circle with a given radius or build on the basis of a given side. The sequence of actions was described in the "Elements" of Euclid about 300 years BC. In any case, we need a pair of compasses and a ruler. Let's consider a method of construction with the help of a given circle.

1. Select an arbitrary radius and draw a circle, marking its center with the point O.

2. On the circle line, select the point that will serve as one of the vertices of our pentagon. Let this be the point A. Join the points O and A by a straight line segment.

3. Draw a straight line through the point O perpendicular to the straight line OA. Point the intersection of this line with the circle line, as point B.

4. At the middle of the distance between points O and B, construct the point C

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

6. Construct a circle passing through D whose center is in A. The points of its intersection with the original circle must be designated by the points E and F.

7. Now construct a circle whose center is in E. Make it necessary so that it passes through A. Its other intersection of the original circle should be designated by the point G.

8. Finally, construct a circle through A with the center at point F. Mark another point of intersection of the original circle by point H.

9. Now we only need to connect the vertices A, E, G, H, F. Our regular pentagon will be ready!