A circular fountain has a diameter of 12 meters and is surrounded by a paved walkway that … — Armed Services Vocational Aptitude Battery (ASVAB)

A circular fountain has a diameter of $12$ meters and is surrounded by a paved walkway that is $3$ meters wide. Which of the following represents the area of the walkway in square meters?

A
$180\pi$
B
$81\pi$
$45\pi$ Answer
D
$6\pi$

Answer

The correct answer is

45\pi

The correct answer is found by subtracting the area of the inner circle from the area of the larger outer circle. The inner radius is

6

meters (half of the

12

-meter diameter), giving an area of

36\pi

. The outer radius is

9

meters (

6

meters plus the

3

-meter walkway), giving a total area of

81\pi

. The difference is

81\pi - 36\pi = 45\pi

Step-by-Step Solution

Find the radius of the inner fountain.

12 \div 2 = 6

meters.

The area formula for a circle requires the radius, which is half of the diameter.

Calculate the area of the fountain.

Area =

\pi \times 6^2 = 36\pi

square meters.

This establishes the area of the space not covered by the walkway.

Find the radius of the total circular area (fountain plus walkway).

6 + 3 = 9

meters.

The walkway adds

3

meters to the radius of the inner circle.

Calculate the total area.

Area =

\pi \times 9^2 = 81\pi

square meters.

This gives the combined area of both the fountain and the walkway.

Subtract the fountain's area from the total area.

81\pi - 36\pi = 45\pi

square meters.

Subtracting the inner area isolates the area of just the paved walkway.

Key Concept

To find the area of a ring (annulus), calculate the area of the outer circle and subtract the area of the inner circle.

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