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Circles: Area

A circle is the set of all points in a plane at a given distance (called the radius ) from a given point (called the center.)

A line segment connecting two points on the circle and going through the center is called a diameter of the circle.

Circle showing radius and diameter

Clearly, if $d$ represents the length of a diameter and $r$ represents the length of a radius, then $d = 2 r$ .

The circumference $C$ of a circle is the distance around the outside. For any circle, this length is related to the radius $r$ by the equation

$C = 2 π r$

where $π$ (pronounced " pi ") is an irrational constant approximately equal to $3.14$ .

The area of a circle is given the formula

$A = π r^{2}$ .

Example 1:

What is the area of a circular table with diameter $6$ ft?

Given that, the diameter of the circular table is $6$ ft.

So, the radius of the circular table is half of the diameter. That is, radius is $3$ ft.

Use the formula for area of a circle, $A = π r^{2}$ , where $r$ is the radius of the circle.

Substitute $3$ for $r$ in the formula.

$\begin{array}{l} A = π {(3)}^{2} \\ = π (9) \\ \approx 28.26 \end{array}$

Therefore, the area of the circular table is about $28.26 {ft}^{2}$

Example 2:

What is the area of the shaded region in the figure shown?

Math diagram

It's clearly marked that the larger circle has a radius $11$ cm. So, its area is

$\begin{array}{l} A_{large circle} = π r^{2} \\ = π {(11)}^{2} \\ = 121 π \\ \approx 379.94 {cm}^{2} \end{array}$

What about the smaller circle? Well, the distance from the center to edge is $11$ cm; the diameter of the smaller circle is $11 - 4 = 7$ cm. So, the radius of the smaller circle is $3.5$ cm.

Math diagram

$\begin{array}{l} A_{small circle} = π r^{2} \\ = π {(3.5)}^{2} \\ = 12.25 π \\ \approx 39.47 {cm}^{2} \end{array}$

To get the area of the shaded region, subtract the area of the smaller circle from the area of the larger circle.

$379.94 - 38.47 = 341.47$

Therefore, the area of the shaded region is about $341.47 {cm}^{2}$ .