A circular stained-glass window panel has a radius of 15 centimeters. A specific red secto… — ACT

A circular stained-glass window panel has a radius of $15$ centimeters. A specific red sector within the panel has an outer arc length of $6\pi$ centimeters. What is the area, in square centimeters, of this red sector?

A
$15\pi$
B
$30\pi$
$45\pi$ Answer
D
$90\pi$
E
$225\pi$

Answer

45\pi

square centimeters

The total circumference of the circle is

2\pi(15) = 30\pi

cm. The sector's arc length of

6\pi

cm is

\frac{6\pi}{30\pi} = \frac{1}{5}

of the entire circle. Since the sector area is proportional to the arc length, the area of the sector is

\frac{1}{5}

of the total area. The total area is

\pi(15)^2 = 225\pi

sq cm, so the sector area is

\frac{1}{5}(225\pi) = 45\pi

sq cm. Alternatively, using the sector area formula

A = \frac{1}{2} r s

, we get

A = \frac{1}{2}(15)(6\pi) = 45\pi

sq cm.

Step-by-Step Solution

Calculate the total circumference of the circle.

Circumference

C = 2\pi r = 2\pi(15) = 30\pi

Knowing the total circumference allows us to determine what fraction of the circle the arc represents.

Determine the fractional portion of the circle corresponding to the arc.

Fraction

= \frac{\text{Arc Length}}{\text{Circumference}} = \frac{6\pi}{30\pi} = \frac{1}{5}

The central angle ratio is equivalent to the ratio of arc length to total circumference.

Calculate the total area of the circle.

Total Area

A_{\text{total}} = \pi r^2 = \pi (15)^2 = 225\pi

^2

The sector area will be the same fraction of the total area as the arc length is of the circumference.

Multiply the fraction by the total area to find the sector area.

Sector Area

= \frac{1}{5} \times 225\pi = 45\pi

^2

Applying the fraction

\frac{1}{5}

to the full circle area yields the targeted sector area.

Key Concept

Relationship between Arc Length and Sector Area in Circle Geometry

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic