Draw some regular polygons along with their circumscribed circles. How can you rewrite the area of the regular polygon?
See solution.
Practice makes perfect
The area of a regular polygon with n sides is given by the following formula, where a is the apothem and P is the perimeter of the polygon.
A = 1/2 a P
Next, let's draw a regular polygon and increase the value of n — that is, we will increase the number of sides.
As we have seen, as the number of sides increases, the polygon becomes more and more circular. Therefore, we can say that as n approaches infinity, the polygon approaches a circle (the circumscribed circle).
In consequence, the perimeter P of the polygon approaches the circumference of a circle — P approaches 2π r, where r is the radius of the circumscribed circle.
P ⟶ 2π r as n→ ∞
The apothem of a regular polygon is the distance from the center of the any side. As n increases, the sides become shorter and shorter, which implies that a approaches more and more to the distance between the center and the vertices of the polygon.
This implies that the apothem a approaches to the radius of the polygon which is the same as the radius of the circumscribed circle.
a ⟶ r as n→ ∞
From the three answers, we can conclude that, as n approaches to infinity, the area of the regular polygon approaches the area of the circumscribed circle.
A = 1/2 a P n→∞⟶ 1/2r(2π r) = π r^2
