A sector of a circle has a central angle of \frac{3\pi}{4} radians and an area of 3\pi squ… — ACT

A sector of a circle has a central angle of $\frac{3\pi}{4}$ radians and an area of $3\pi$ square inches. What is the perimeter, in inches, of the sector?

A
$\frac{3\pi\sqrt{2}}{2}$
B
$2\sqrt{2} + \frac{3\pi\sqrt{2}}{2}$
$4\sqrt{2} + \frac{3\pi\sqrt{2}}{2}$ Answer
D
$4 + \frac{3\pi}{2}$
E
$8 + 3\pi$

Answer

The perimeter of the sector is

4\sqrt{2} + \frac{3\pi\sqrt{2}}{2}

inches.

The area of a sector is given by

A = \frac{1}{2} r^2 \theta

. Substituting the given area

3\pi

and the central angle

\frac{3\pi}{4}

, we get

3\pi = \frac{3\pi}{8} r^2

, which simplifies to

r^2 = 8

and

r = 2\sqrt{2}

. The perimeter of the sector consists of the curved arc length

s = r\theta

plus the two straight radii of the sector (

2r

). Substituting the values yields a perimeter of

2(2\sqrt{2}) + (2\sqrt{2})\left(\frac{3\pi}{4}\right) = 4\sqrt{2} + \frac{3\pi\sqrt{2}}{2}

Step-by-Step Solution

Use the sector area formula in radians,

A = \frac{1}{2} r^2 \theta

, to solve for the radius

r

of the circle.

3\pi = \frac{1}{2} r^2 \left(\frac{3\pi}{4}\right) \implies 3\pi = \frac{3\pi}{8} r^2 \implies r^2 = 8 \implies r = 2\sqrt{2}

inches.

Finding the radius of the circle is necessary to calculate both the arc length and the straight boundary segments of the sector.

Calculate the arc length

s

of the sector using the formula

s = r\theta

s = (2\sqrt{2})\left(\frac{3\pi}{4}\right) = \frac{3\pi\sqrt{2}}{2}

inches.

The arc length represents the curved boundary of the sector.

Compute the total perimeter of the sector by adding the arc length to the two boundary radii:

P = 2r + s

P = 2(2\sqrt{2}) + \frac{3\pi\sqrt{2}}{2} = 4\sqrt{2} + \frac{3\pi\sqrt{2}}{2}

inches.

The perimeter of a sector is the sum of the curved arc length and the two straight radial boundary segments.

Key Concept

Calculating the perimeter of a circular sector using its area and central angle in radians.

Estimated Time:1m 35s

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