Area of a Circle

1. What is the Area of a Circle?

The area of a circle is the amount of space enclosed within its boundary (circumference). It tells you how much surface the circle covers.

For example, the area of a circular field shows how much land it covers.

The area depends only on the radius of the circle.

A = \pi r^2

Where:

The number \(\pi\) appears because it relates the curved boundary of a circle to its diameter.

When we break a circle into many narrow slices and rearrange them (like a zig-zag pattern), they form a shape similar to a rectangle.

Imagine cutting the circle into many equal wedges:

So the area becomes:

A = \text{base} \times \text{height} = \pi r \times r = \pi r^2

If diameter \(d = 2r\), then the area can also be written as:

A = \pi \left(\dfrac{d}{2}\right)^2 = \dfrac{\pi d^2}{4}

Area is always expressed in square units:

Here are simple examples to understand the formula better:

Find the area of a circle with radius \(r = 10\text{ cm}\).

A = \pi r^2 = \pi \times 10^2 = 100\pi\text{ cm}^2

Find the area of a circle of diameter \(14\text{ cm}\).

A = \dfrac{\pi d^2}{4} = \dfrac{\pi \times 14^2}{4} = 49\pi\text{ cm}^2

The area formula is widely used in: