Section 13.3: Areas of Circles

Property

A circle is defined as all points that are the same distance from a center point. The radius ($r$) is any line segment from the center of the circle to a point on the circle. The diameter ($d$) is any line segment that passes through the center and connects two points on the circle, where $d = 2r$.

Examples

• When you drop a pebble in a pond, the ripples form circles. Every point on a single ripple is the same distance from where the pebble hit the water (the center).
• A pizza has a circular shape. If the pizza has a radius of $r = 6$ inches, then its diameter is $d = 2 \times 6 = 12$ inches.
• A bicycle wheel is circular. The distance from the center hub to the edge of the tire is the radius, while the distance across the entire wheel through the center is the diameter.

Explanation

Understanding the radius and diameter of a circle is essential for calculating its area. The radius is the key measurement we use in the area formula, while the diameter helps us find the radius when needed. These definitions form the foundation for working with circle area problems.

Property

The ratio of the area of a circle and the square of its radius is a constant, and that constant is the area of a circle of radius 1 unit, and is designated by the Greek letter $\pi$. Thus we get this formula for the area $A$ of a circle in terms of its radius $r$:

Examples

• A circle has a radius of $5$ cm. Its area is $A = \pi (5^2) = 25\pi$ cm$^2$.
• A pizza has an area of $100\pi$ square inches. To find its radius, we solve $100\pi = \pi r^2$, which gives $r^2 = 100$, so the radius is $10$ inches.
• A circular garden has a diameter of $12$ meters. The radius is half the diameter, so $r = 6$ m. The area is $A = \pi (6^2) = 36\pi$ m$^2$.

Explanation

The area of a circle grows with the square of its radius. This means if you double the radius, the area doesn't just double, it becomes four times larger! The special number $\pi$ is the constant that links them.

Property

By imagining a circle's area as being rearranged into a triangle, we can derive the area formula in a visual way.
The triangle's base would be the distance around the circle and its height would be the radius ($r$).
This gives us the formula for the area of the circle:

Since the distance around a circle is $2\pi r$, this becomes: