A circle has a radius of 6. A central angle of 60^\circ intercepts an arc on the circle. W… — GRE General Test

A circle has a radius of $6$ . A central angle of $60^\circ$ intercepts an arc on the circle. What is the perimeter of the sector defined by this central angle?

A
$2\pi$
B
$6\pi$
C
$2\pi + 6$
$2\pi + 12$ Answer
E
$12\pi$

Answer

2\pi + 12

The sector's perimeter includes the curved arc length and the two straight radii bounding it. The central angle of

60^\circ

represents

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

of the full circle. The arc length is

\frac{1}{6} \times 2\pi(6) = 2\pi

. Adding the two radii of length

6

gives

2\pi + 6 + 6 = 2\pi + 12

Step-by-Step Solution

Calculate the arc length of the sector

Arc length =

2\pi

The arc length formula is

\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r

. Substituting

\theta = 60^\circ

and

r = 6

yields

\frac{60}{360} \times 12\pi = \frac{1}{6} \times 12\pi = 2\pi

Calculate the perimeter of the sector by adding the arc length to the two radii

Perimeter =

2\pi + 12

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

2r

). Thus,

\text{Perimeter} = 2\pi + 2(6) = 2\pi + 12

Key Concept

Perimeter of a sector equals arc length plus twice the radius (

L + 2r

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