A section of a circular garden is enclosed by two radii and an outer arc, forming a circul… — ACT

A section of a circular garden is enclosed by two radii and an outer arc, forming a circular sector with a central angle of $60^\circ$ . If the total perimeter of this sector is $12 + 2\pi$ meters, what is the area of the sector, in square meters?

A
$2\pi$
B
$3\pi$
$6\pi$ Answer
D
$12\pi$
E
$36\pi$

Answer

The area of the sector is

6\pi

square meters.

The perimeter of a circular sector is the sum of its two straight edges (radii) and its curved edge (arc length):

\text{Perimeter} = 2r + \frac{\theta}{360^\circ} \cdot 2\pi r

. Substituting

\theta = 60^\circ

gives

2r + \frac{\pi r}{3} = 12 + 2\pi

. Matching corresponding terms yields

2r = 12

, so

r = 6

meters. The sector area is then

\frac{60^\circ}{360^\circ} \cdot \pi (6)^2 = 6\pi

square meters.

Step-by-Step Solution

Express the arc length

s

in terms of the radius

r

s = \frac{60^\circ}{360^\circ} \cdot 2\pi r = \frac{1}{6} \cdot 2\pi r = \frac{\pi r}{3}

meters.

The arc length of a sector with central angle

\theta

in degrees is given by

s = \frac{\theta}{360^\circ} \cdot 2\pi r

Set up and solve the equation for the sector's total perimeter to find radius

r

Perimeter

= 2r + s = 2r + \frac{\pi r}{3} = 12 + 2\pi

. Equating integer and

\pi

components gives

2r = 12 \implies r = 6

meters.

The perimeter of a sector consists of the two bounding radii plus the arc length.

Calculate the area of the sector using

r = 6

meters.

Sector Area

= \frac{60^\circ}{360^\circ} \cdot \pi r^2 = \frac{1}{6} \cdot \pi (6)^2 = \frac{36\pi}{6} = 6\pi

square meters.

The area of a circular sector is given by

A = \frac{\theta}{360^\circ} \cdot \pi r^2

Key Concept

Perimeter and Area of a Circular Sector

Estimated Time:1m 30s

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