Regular pentagon: the necessary minimum of information

2024 Author: Angel Austin | [email protected]. Last modified: 2023-12-17 05:14

Ozhegov's Explanatory Dictionary states that a pentagon is a geometric figure bounded by five intersecting straight lines forming five internal angles, as well as any object of a similar shape. If a given polygon has the same sides and angles, then it is called a regular (pentagon).

What is interesting about a regular pentagon?

It was in this form that the well-known building of the United States Department of Defense was built. Of the voluminous regular polyhedra, only the dodecahedron has pentagon-shaped faces. And in nature, crystals are completely absent, the faces of which would resemble a regular pentagon. In addition, this figure is a polygon with a minimum number of corners that cannot be used to tile an area. Only a pentagon has the same number of diagonals as its sides. Agree, it's interesting!

Basic properties and formulas

Using the formulas forarbitrary 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 interior angles is 540°.
• The ratio of the diagonal to the side is (1+√5) /2, that is, the "golden section" (approximately 1, 618).
• The length of the side that a regular pentagon has can be calculated using one of three formulas, depending on which parameter is already known:
• if a circle is circumscribed around it and its radius R is known, then a=2Rsin (α/2)=2Rsin(72°/2) ≈1, 1756R;
• in the case when a circle with radius r is inscribed in a regular pentagon, a=2rtg(α/2)=2rtg(α/2) ≈ 1, 453r;
• 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 a regular pentagon is determined, again, depending on what parameter we know:
• if there is an inscribed or circumscribed circle, then one of two formulas is used:

S=(nar)/2=2, 5ar or S=(nR²sin α)/2 ≈ 2, 3776R²;

the area can also be determined by knowing only the length of the side a:

S=(5a²tg54°)/4 ≈ 1, 7205 a².

Regular pentagon: construction

This geometric figure can be built in different ways. For example, inscribe it in a circle with a given radius, or build it on the basis of a given lateral side. The sequence of actions was described in Euclid's Elements around 300 BC. In any case, we need a compass and a ruler. Consider the construction method using a given circle.

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

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

3. Draw a line through point O perpendicular to line OA. Designate the intersection of this line with the line of the circle 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 line OB (it will be inside the very first circle) will be point D.

6. Construct a circle passing through D, the center of which will be in A. The places of its intersection with the original circle must be marked with points E and F.

7. Now construct a circle, the center of which will be in E. You need to do this so that it passes through A. Its other intersection of the original circle must be indicated by point G.

8. Finally, draw a circle through A centered at point F. Mark another intersection of the original circle with point H.

9. Now leftjust connect vertices A, E, G, H, F. Our regular pentagon will be ready!