Use Limit to find the area and circumference of a circle

Use Limit to find the area and circumference of a circle

A = r² , C = 2r

Yue Kwok Choy

1. Prove that : .

Proof:

Case 1 :  > 0 .

As in the diagram, AC is the tangent to the circle.

AC = r tan  , where r is the radius of the circle.

Area of OAB < area of sector OAB < area of OAC

(on dividing by sin )

Since .

Case 2 :  < 0 , put  = –  , then  > 0 .

Combine Case 1 and Case2 , we have .

2.

Proof: .

3. Area of an n-sided regular polygon inscribed in a circle with radius r = .

Proof: Exercise

4. Area of a circle with radius r = r² .

Proof: Area of a circle =

5. Perimeter of an n-sided regular polygon inscribed in a circle with radius r = .

Proof: Exercise

6. Circumference of a circle with radius r = 2r .

Proof: Circumference of a circle =