A decorative circular stained-glass window has a radius of 10 inches. A specific colored s… — ACT

A decorative circular stained-glass window has a radius of $10$ inches. A specific colored section of the window forms a sector with a central angle measuring $108^\circ$ . What is the total perimeter, in inches, of this stained-glass sector?

A
$6\pi$
B
$3\pi + 20$
C
$6\pi + 10$
$6\pi + 20$ Answer
E
$30\pi + 20$

Answer

The total perimeter of the sector is

6\pi + 20

inches.

The arc length for a central angle of

108^\circ

in a circle of radius

10

\frac{108}{360} \times 2\pi(10) = 6\pi

inches. Adding the two bounding radii (

10 + 10 = 20

) yields the full sector perimeter of

6\pi + 20

inches.

Step-by-Step Solution

Calculate the arc length of the sector

Arc length =

\frac{108^\circ}{360^\circ} \cdot 2\pi(10) = \frac{3}{10} \cdot 20\pi = 6\pi

inches.

The curved boundary of a sector is a fraction of the circle's full circumference defined by the central angle ratio.

Calculate the total perimeter of the sector

Perimeter = Arc length +

2 \cdot \text{radius} = 6\pi + 2(10) = 6\pi + 20

inches.

A sector's perimeter consists of its outer curved arc plus its two straight radial edges.

Key Concept

Perimeter of a Sector

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