Ozhegov’s Explanatory Dictionary states that a pentagon is a geometric figure bounded by five intersecting straight lines forming five internal corners, as well as any object of a similar shape. If a given polygon has all sides and angles the same, then it is called regular (pentagon).
Why is a regular pentagon interesting?
It was in this form that the well-known building of the United States Department of Defense was built by everyone. Of the volumetric regular polyhedra, only the dodecahedron has faces in the form of a pentagon. But in nature there are completely no crystals whose faces would resemble a regular pentagon. In addition, this figure is a polygon with a minimum number of angles that cannot be tiled with an area. Only in a pentagon the number of diagonals coincides with the number of its sides. Agree, this is interesting!
Basic properties and formulas
Using the formulas for an arbitrary regular polygon, you can determine all the necessary parameters that the pentagon has.
- Central angle α = 360 / n = 360/5 = 72 °.
- Internal angle β = 180 ° * (n-2) / n = 180 ° * 3/5 = 108 °. Accordingly, the sum of the internal angles is 540 °.
- The ratio of the diagonal to the side is (1 + √5) / 2, that is, the "golden ratio" (approximately 1.618).
- The length of the side that the regular pentagon has can be calculated using one of three formulas, depending on which parameter is already known:
- if a circle is described 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 the radii the diagonal 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 we know:
- 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 by knowing only the length of the side a:
S = (5 * a 2 * tg54 °) / 4 ≈ 1.7205 * a 2 .
Regular pentagon: building
This geometric shape can be built in different ways. For example, fit it into a circle with a given radius or build on the basis of a given side. The sequence of actions was described back in the "Beginnings" of Euclid approximately 300 years BC In any case, we need a pair of compasses and a ruler. Consider the construction method using 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 a point that will serve as one of the vertices of our pentagon. Let it be point A. Connect the points O and A with a straight line.
3. Draw a line through point O perpendicular to line OA. The intersection of this line with the circle line is designated as point B.
4. In the middle of the distance between points O and B, build 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 point D.
6. Construct a circle passing through D whose center will be in A. The places of its intersection with the original circle must be denoted by points E and F.
7. Now build a circle whose center will be in E. You need to do this so that it passes through A. Its other intersection of the original circle must be denoted by G.
8. Finally, draw a circle through A centered at F. Designate another intersection of the original circle with H.
9. Now it remains only to connect the vertices A, E, G, H, F. Our regular pentagon will be ready!