Area of regular polygons

If we want to know the approximate area of a circle by increasing the number of sides of the regular polygon inside a circle we are going to need to know how to calculate the area of a regular polygon.

To calculate the area of a regular pentagon, we can start by dividing it up into triangles.

A regular pentagon is made up of five congruent triangles. If we can find the area of one of the triangles, we can find the area of the pentagon.

The circle that the regular pentagon is inside has a radius of 10cm. The dotted sides of the triangles are 10cm as well.

What else do we know about the triangles?

We know that there are 360° in a circle.

The angle at the centre of the circle for each triangle is the same. Each angle is a fifth of 360. The blue triangle, for example has an angle of 360/5 = 72.

The angle at the centre for each triangle is 72°

Let’s look at one of the five triangles, the blue one in the pentagon above.

To find the area of this triangle, we need to know its perpendicular height.

We know the angle between the perpendicular dotted line and the side labelled with the length is half of 72 because the perpendicular bisects the 72 degree angle.

Let’s look at one of the right angled triangles created by the perpendicular.

We know that the angles in a triangle add up to 180 degrees.

The remaining angle is 180 – 36 – 90 = 54 degrees.

We could have got the angles in the right angled triangle another way. We could have noticed that the five triangles that we have created from the pentagon are all isosceles triangles.

Isosceles triangles have two sides that are the same length and where these two sides join the base they make two angles of the same size. As the angles of a triangle add up to 180° then the two angles must be

(180 – 72) ÷ 2

= 108 ÷ 2

= 54 degrees

To calculate the perpendicular height, however, we only need the length of one side and any angle because we can use our trigonometric ratios.

Let’s work with the information here.

We have the length of the hypotenuse.

We want to know the length of the adjacent side to the 36° angle

We know cos θ = adjacent / hypotenuse

cos θ = A / H

cos 36 = A / 10

A = 10 * cos 36

A = 8.09

We could also have worked with the information here.

Here we have the hypotenuse but want to know the length of the opposite.

sin θ = opposite / hypotenuse

sin θ = O / H

sin 54 = O / 10

O = 10 * sin 54

O = 8.09

Now we have the height of the triangle, but we still need the length of the base to work out the area.

We could work out the length of base in the triangle using the other angle instead of this one but here we want to know the length of the adjacent side to the 54° angle.

cos θ = adjacent / hypotenuse

cos θ = A / H

cos 54 = A /10

A = 10 * cos 54

A= 5.88

We could also have used the sine function or Pythagoras’ Theorem to work out the length of the third side of the right-angled triangle.