A sector of a circle with radius r inches has a central angle measuring 60^\circ. The rati… — ACT

A sector of a circle with radius $r$ inches has a central angle measuring $60^\circ$ . The ratio of the area of the sector (in square inches) to the perimeter of the sector (in inches) is $3:1$ . What is the radius, $r$ , of the circle, in inches?

$\frac{36 + 6\pi}{\pi}$ Answer
B
$6$
C
$\frac{12 + 2\pi}{3\pi}$
D
$\frac{18 + 6\pi}{\pi}$
E
$\frac{18 + 3\pi}{\pi}$

Answer

The radius of the circle is

\frac{36 + 6\pi}{\pi}

inches.

The sector area is

\frac{1}{6}\pi r^2

and the total sector perimeter (arc length plus two radii) is

2r + \frac{\pi r}{3}

. Setting the area equal to 3 times the perimeter gives

\frac{1}{6}\pi r^2 = 6r + \pi r

. Dividing by non-zero

r

gives

\frac{1}{6}\pi r = 6 + \pi

, which yields

r = \frac{36 + 6\pi}{\pi}

Step-by-Step Solution

Express the sector area in terms of radius

r

Sector Area

= \frac{60^\circ}{360^\circ} \times \pi r^2 = \frac{1}{6}\pi r^2

The area of a sector with central angle

\theta

in degrees is

\frac{\theta}{360^\circ}\pi r^2

Express the sector arc length and total sector perimeter in terms of

r

Arc Length

= \frac{60^\circ}{360^\circ} \times 2\pi r = \frac{\pi r}{3}

; Sector Perimeter

= 2r + \frac{\pi r}{3}

The total perimeter of a sector includes the curved arc length plus the two straight radii that enclose it.

Set up the equation using the given ratio of Area to Perimeter (

3:1

\frac{\frac{1}{6}\pi r^2}{2r + \frac{\pi r}{3}} = 3 \implies \frac{1}{6}\pi r^2 = 3\left(2r + \frac{\pi r}{3}\right)

A ratio of

3:1

means Area

= 3 \times \text{Perimeter}

Solve the equation for

r

\frac{1}{6}\pi r^2 = 6r + \pi r \implies \frac{1}{6}\pi r = 6 + \pi \implies r = \frac{6(6 + \pi)}{\pi} = \frac{36 + 6\pi}{\pi}

Dividing both sides by

r

(since

r > 0

) simplifies the quadratic relationship to a linear equation in

r

Key Concept

Calculating sector area, arc length, and sector perimeter using central angle ratios.

Estimated Time:2m 0s

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