Area of a Circle – Proof | Geometry | Maths | Tutorial

You may not know this but there is no way of calculating the area of a circle exactly. The area of a circle is calculated by splitting the circle into many right-angle triangles and then calculating the area of these triangles. As you know triangles have straight lines and circles are curved therefore there is a margin of error for the area between the edge of the circle and the side of the triangle.

In this tutorial we will look at how to derive the formula for the area of a circle. For instructions on how to use the formula read article “How to Calculate the Area of a Circle”

 Illustration

 Things to know

A = Area

π(pi) = 3.14159….

r = Radius (Half of the diameter)

c = Circumference

Proof

Step 1: Area of a right-angle triangle

¹⁄₂ x Base x Height = ¹⁄₂ x ^c⁄₆ x r

Note: There are 16 triangles in this circle so we divide the circumference by 16 to get the value of the height of the triangle.

Step 2: Area of circle

We need to multiply the equation by 16 to get the area of all 16 triangles that make up the circle and simplify the equation.

16 x ¹⁄₂ x ^c⁄₁₆ x r = ¹⁄₂ x c x r

Note: I have used 16 triangles as per the illustration above however you can use any number of triangles. This number is irrelevant to the proof as it is cancelled out when the equation is simplified.

Step 3: Substitute the value of c

The formula for the circumference (c) of a circle is C = 2πr. We can substitute the value of c to the equation.

¹⁄₂ (2 π r) x r = π r²