Lesson 40: Area of a Circle

New Concept

The area of a circle is $\pi$ times the area of a square on its radius. This relationship is expressed with the formula:

What’s next

This card is just the start. Next, you’ll tackle worked examples for finding the area of full circles and even parts of circles, called sectors.

Property

The area of a circle is $\pi$ times the area of a square built on the radius. The formula is:

Examples

For a radius of 5 cm: $A = \pi(5 \text{ cm})^2 = 25\pi \text{ cm}^2$.
For a diameter of 6 feet, the radius is 3 feet: $A = \pi(3 \text{ ft})^2 \approx 3.14(9 \text{ ft}^2) = 28.26 \text{ ft}^2$.

Explanation

Think of the area as being just a little more than three squares made from the circle's radius! The magic number $\pi$ (pi) is the special ingredient that tells us exactly how many of those squares can fit inside the circle. It’s always the same amount for any circle.

Property

A sector is a portion of a circle's interior enclosed by two radii and an arc. The area is a fraction of the circle's total area, determined by the central angle:

Examples

For a $45^\circ$ sector with a 4 cm radius: Area = $\frac{45^\circ}{360^\circ} \cdot \pi(4 \text{ cm})^2 = \frac{1}{8} \cdot 16\pi \text{ cm}^2 = 2\pi \text{ cm}^2$.
A semicircle (a $180^\circ$ sector) with a 10m radius: Area = $\frac{1}{2} \cdot \pi(10 \text{ m})^2 = 50\pi \text{ m}^2$.

Explanation

Imagine a sector as a perfect slice of pizza! The angle at the pointy end tells you exactly what fraction of the whole pizza you get. A $90^\circ$ angle means you get a quarter of the pie, while a $180^\circ$ angle gets you a delicious half!