Regular pentagon: minimum information

Ozhegova Explanatory Dictionary states that the pentagon is a geometric figure bounded by five intersecting lines that form five interior angles, as well as any object of similar shape.If the given polygon all sides and angles are the same, it is called a right (the Pentagon).

What is interesting regular pentagon?

in this form was built the well-known building of the Ministry of Defense of the United States.From the volume of regular polyhedra a dodecahedron has the edge in the form of pentagon.In nature there are no crystals at all, the faces that resemble to a regular pentagon.Moreover, this figure is a polygon with the minimum number of angles, which can not be tiled area.Only the number of diagonals of a pentagon coincides with the number of its sides.Agree, it's fun!

Basic properties and formulas

Using the formulas for any regular polygon, you can define all the necessary parameters, which is the Pentagon.

  • central angle α = 360 / n = 360/5 = 72 °.
  • inner angle β = 180 ° * (n-2
    ) / n = 3/5 * 180 ° = 108 °.Accordingly, the sum of the interior angles is 540 °.
  • ratio of the diagonal to the side is (1 + √5) / 2, that is, the "golden section" (approximately 1,618).
  • length party who is a regular pentagon can be calculated by one of the three formulas, depending on which option is already known:
  • if around the circumscribed circle and is known for its radius R, then a = 2 * R* sin (α / 2) = 2 * R * sin (72 ° / 2) * R ≈1,1756;
  • when c circle radius r inscribed in a regular pentagon, a = 2 * r * tg (α / 2) = 2 * r * tg (α / 2) ≈ 1,453 * r;
  • happens that instead of radii known value of the diagonal D, then the direction is determined as follows: a ≈ D / 1,618.
  • area of ​​a regular pentagon is determined, again, depending on which parameter we know:
  • if there is inscribed or circumscribed circle, then use one of two formulas:

S = (n * a * r) / 2 = 2,5 * a * r or S = (n * R2 * sin α) / 2 ≈ 2.3776 * R2;

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

S = (5 * a2 * tg54 °) / 4 ≈ 1,7205 * a2.

correct pentagon: the construction

This geometric shape is possible to build differently.For example, to write it into a circle with a predetermined radius based on a predetermined build side.The sequence of actions has been described in the "Elements" of Euclid around 300 BCIn any case, we need a compass and a ruler.Consider using a method of constructing a given circle.

1. Select an arbitrary radius and draw a circle, denoting its center point O.

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

3. Draw a line through the point perpendicular to the line OA.Place the intersection of this line with the line of the circle mark as point B.

4. In the middle of the distance between the points O and B build point C.

5. Now draw a circle whose center is at the point, and that will pass through point A. place of its intersection with the line OB (it will appear within the first circle) will point D.

6. Construct a circle through D, the center of which will be in the A. Designated its intersection with the original circle is necessary to designate the points E and F.

7. Now build a circle whose center is in E. To do this it is necessary so that it passes through A. It is another point of intersection of the original circle is necessary to designate point G.

8. Finally, draw a circle through the center of A at point F. Designate a different location point of intersection of the original circle H.

9. Now you only have to connect the top of the A, E, G, H, F. Our regular pentagon will be ready!