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How to find the area of an annulus

The rule: subtract the area of the hole

An annulus is the ring between two circles with the same center. If $R$ is the outer radius and $r$ is the inner radius, subtract the small circle from the large one:

A = π R^{2} - π r^{2} = π (R^{2} - r^{2})

Do not use $π (R - r)^{2}$ . The area depends on the difference of the squares, not the square of the difference.

Problem

A ring-shaped region has inner radius $6 cm$ and outer radius $9 cm$ . Find the shaded area. Use $3.14$ for $π$ and do not round.

The shaded ring is the area of the outer circle minus the area of the inner circle.

Answer

141.3 cm^{2}

Checked by computing the two circle areas separately and subtracting.

Step-by-step solution

1. Area of the outer circle

The outer radius is $R = 9 cm$ :

A_{outer} = 3.14 \cdot 9^{2} = 3.14 \cdot 81 = 254.34

2. Area of the inner circle

The inner radius is $r = 6 cm$ :

A_{inner} = 3.14 \cdot 6^{2} = 3.14 \cdot 36 = 113.04

3. Subtract the hole

A_{ring} = 254.34 - 113.04 = 141.30

So the shaded area is

A = 141.3 cm^{2}

One-line method

The same calculation can be written directly as

A = 3.14 (9^{2} - 6^{2}) = 3.14 (81 - 36) = 3.14 \cdot 45 = 141.3 cm^{2}

The units are square centimeters because this is an area.

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