Lesson Area of a regular n-sided polygon via the radius of the circumscribed circle

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Area of a regular n-sided polygon via the radius of the circumscribed circle


In this lesson you will learn how to calculate the area of a regular  n-sided polygon via the radius of the circumscribed circle.

Theorem

The area  S  of a regular  n-sided polygon equals

S = n.R%5E2.cos%28pi%2Fn%29.sin%28pi%2Fn%29 = 1%2F2.n.R%5E2.sin%282pi%2Fn%29,          (1)

where  R  is the radius of the circle circumscribed about the polygon.              

Proof

(Figure 1a  and  Figure 1b  show a regular pentagon and a regular
hexagon along with their circumscribed circles as the examples).

For the given regular  n-sided polygon,  let us draw the circumscribed
circle.  You know from the lesson  Regular polygons  that it is always
possible to do so.  The center of this circle coincides with the center



  Figure 1a.  To the  Theorem 1        



  Figure 1b.  To the  Theorem 1
of the given regular polygon.  Let  R  be the radius of this circle.

Connect the center of the polygon with its vertices by the straight segments.  These straight segments divide the interior of the polygon in  n  isosceled congruent triangles.
In each of these triangles,  the altitude is the median and the angle bisector simultaneously.  Therefore,  the measure of the altitude is  R%2Acos%282pi%2F%282n%29%29 = R%2Acos%28pi%2Fn%29.  The base of each triangle is  2R%2Asin%28pi%2Fn%29  in accordance with the lesson  The side length of a regular polygon via the radius of the circumscribed circle.  Hence,  the area of each isosceles triangle is   R%5E2%2Acos%28pi%2Fn%29%2Asin%28pi%2Fn%29,  and the area of the entire polygon is  nR%5E2.cos%28pi%2Fn%29.sin%28pi%2Fn%29.  It proves the first half of the formula  (1).

To prove the second half of the formula  (1),  use the formula  sin%28beta%29%2Acos%28beta%29 = %28sin%282beta%29%29%2F2  from  Trigonometry  to replace  cos%28pi%2Fn%29.sin%28pi%2Fn%29  by  1%2F2.sin%282pi%2Fn%29.
Or,  alternatively,  simply use the formula  S = 1%2F2.a%2Ab%2Asin%28alpha%29  for the area of each isosceles triangle with  a = R,  b = R  and  alpha = 2pi%2Fn.


Example 1

The area of a regular triangle equals

S = 3.R%5E2.cos(60°).sin(60°) = 3.R%5E2.1%2F2.sqrt%283%29%2F2 = 3sqrt%283%29%2F4R%5E2,

where  R  is the radius of the circumscribed circle.


Example 2

The area of a square equals

S = 4.R%5E2.cos(45°).sin(45°) = 4.R%5E2.sqrt%282%29%2F2.sqrt%282%29%2F2 = 2R%5E2,

where  R  is the radius of the circumscribed circle.


Example 3

The area of a regular hexagon equals

S = 1%2F26.R%5E2.sin(60°) = 3.R%5E2.sqrt%283%29%2F2 = 3sqrt%283%29%2F2R%5E2,

where  R  is the radius of the circumscribed circle.


Problem 1

Find the area of a regular triangle inscribed in the circle of the radius of  10 cm.

Solution

Use the formula of the  Theorem 1  above.  You have

S = 3.R%5E2.cos(60°).sin(60°) = 3.10%5E2.1%2F2.sqrt%283%29%2F2 = 3sqrt%283%29%2F4.100 = 75.1.732 = 129.90 (approximately).

Answer. The area of the regular triangle is  129.90 cm%5E2  (approximately).


Problem 2

Find the area of a regular pentagon inscribed in the circle of the radius of  10 cm.

Solution

Use the formula of the  Theorem 1  above.  You have

S = 1%2F2.n.R%5E2.sin%282pi%2Fn%29 = 1%2F2.5.100.sin%282pi%2F5%29 = 250.sin(72°) = 250.0.951 = 237.76 cm%5E2  (approximately).

Answer. The area of the regular pentagon is  237.76 cm%5E2  (approximately).
.

Problem 3

Find the area of a regular hexagon inscribed in the circle of the radius of  10 cm.

Solution

Use the formula of the  Theorem 1  above.  You have

S = 1%2F2.n.R%5E2.sin%282pi%2Fn%29 = 1%2F2.6.100.sin%282pi%2F6%29 = 300.sin(60°) = 300.sqrt%283%29%2F2 = 259.81 cm%5E2  (approximately).

Answer. The area of the regular hexagon is  259.81 cm%5E2  (approximately).


My other lessons on the area of polygons in this site are
    - Area of n-sided polygon circumscribed about a circle  and
    - Area of a regular n-sided polygon via the radius of the inscribed circle
under the topic  Area and surface area  of the section  Geometry,  and
    - Solved problems on area of regular polygons
under the topic  Geometry  of the section  Word problems.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link  GEOMETRY - YOUR ONLINE TEXTBOOK.

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