Area of regular polygons part two

If we drew regular polygons with 6, 7, 8, 9, 10 sides then what area would the polygon be?

A pentagon, with five sides, can be split into 5 similar triangles that have an angle at the centre of 360/5 = 72°

A hexagon, with six sides, can be split into 6 similar triangles. Each has an angle of 360/6 = 60° at the centre.

We can see how this pattern continues in the table below

For each angle at the centre of the circle, we are interested in half of the angle because bisecting the angle creates two right angled triangles. From this we can work out the height of the triangle and the length of the base and so we can calculate the area of the triangle and then the area of the polygon.

Let’s say that the radius of the circle that the polygons will be inside of is 10cm.

For each angle θ, we need to calculate the height and the base of the triangle shown.

The formula involving the height is cos θ = height/10

height = 10 cos θ

The formula involving the base is sin θ = base/10

base = 10 sin θ

Then to calculate the area of the isosceles triangle we multiply these together and to calculate the area of the polygon we multiply the area of the isosceles triangle by the number of sides of the polygon.

(add an example)

Let’s add this information to our table and extend the table to include 20, 30 , 40 and 100 sided polygons