A circular metal plate has a radius of 6\text{ inches}. A sector of the plate with a centr… — ACT

A circular metal plate has a radius of $6\text{ inches}$ . A sector of the plate with a central angle of $60^\circ$ is cut out to make a custom spacer. What is the area, in square inches, of the cut-out sector?

A
$2\pi$
B
$\frac{3\pi}{2}$
$6\pi$ Answer
D
$\frac{36\pi}{5}$
E
$36\pi$

Answer

The correct area of the sector is

6\pi

square inches.

The correct answer of

6\pi

is found by calculating the total area of the circular plate using

A = \pi r^2 = \pi (6)^2 = 36\pi

, and then multiplying this by the ratio of the central angle to the total degrees in a circle, which is

\frac{60^\circ}{360^\circ} = \frac{1}{6}

. Thus, the sector area is

\frac{1}{6} \times 36\pi = 6\pi

square inches.

Step-by-Step Solution

Calculate the area of the entire circular plate.

The total area of the circle is

A = \pi (6)^2 = 36\pi

square inches.

Before finding the area of a sector, we need the total area of the circle of which it is a part.

Determine the fraction of the circle represented by the sector.

The fraction is

\frac{60^\circ}{360^\circ} = \frac{1}{6}

A circle contains

360^\circ

, so a central angle of

60^\circ

corresponds to

\frac{60}{360}

of the full circle.

Multiply the total area of the circle by the fraction.

The sector area is

36\pi \times \frac{1}{6} = 6\pi

square inches.

The area of a sector is proportional to its central angle relative to the total angle of a circle.

Key Concept

The area of a sector with radius

r

and central angle

\theta

in degrees is given by the formula

A = \pi r^2 \left(\frac{\theta}{360^\circ}\right)

Alternative Method

Alternatively, since

60^\circ

\frac{1}{6}

of a full

360^\circ

circle, the sector area is simply one-sixth of the total area of the circle (

36\pi

), which gives

6\pi

square inches.

Estimated Time:45s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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