ACT Math: A Challenging Arc Length & Sector Area Problem

One of our more popular posts involves Arc Lengths & Sector Areas for circles. With that in mind, we thought we’d give you a challenging problem (well, really two problems) involving this topic. Give it a try and then scroll down for the answer.

The circle below has its center at O and triangle AOB is equilateral with sides that are each 6 inches.

1.  What is the area, in square inches, of the shaded region?

A.  6π – 3√3
B.  9π – 3√3
C.  6π – 6√3
D.  6π – 9√3
E.  9π – 9√3

2.  What is the perimeter, in inches, of the shaded region?

F.  2π + 6
G.  2π + 18
H.  4π + 6
J.  4π + 18
K.  6π + 6 

When you’re ready, scroll down for the solutions.
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Solutions

1. The area of the shaded region is equal to the area of the triangle subtracted from the area of the sector. Begin by finding each of these areas. To find the area of the sector you can either use the formula for sector area or view the sector as some part of the total area of the circle.

By formula:
A =(1/2)(r2)(θ)
A = (1/2)(62)(π/3) = 6π

(Note: The triangle is equilateral so each of its angles has a degree measure of 60°.  A degree measure of 60° is equivalent to π/3 radians.)

As part of a circle:
Full circle:
A = πr2
A = π(62) = 36π

Sector:
A = (60/360)(36π)
A = (1/6)(36π)
A = 6π

(Note: The angle in the triangle, which is also the central angle of the sector is 60°. Therefore, the sector’s area is 60/360 or 1/6 the area of the full circle.)

To find the triangle’s area, use the formula for the area of an equilateral triangle:

A = (6²√3)/4
A = 36√3/4
A = 9√3

Therefore, the area of the shaded region is equal to 6π – 9√3 and the correct answer is Choice D.

2. The perimeter of the shaded region consists of one side of the triangle and the arc length for the central angle of 60°. The side of the triangle is 6. The arc length can be found using the arc length formula or by thinking of it as some part of the full circumference of the circle.

By formula:
s = r·θ
s = 6(π/3)
s = 2π

As part of the circumference:
Full circumference:
C = 2πr
C = 2π(6)
C = 12π

Arc Length:
s = (60/360)(12π)
s = (1/6)(12π)
s = 2π

So the perimeter of the shaded region is 2π + 6 and the correct answer is Choice F.

If you have questions about this problem or anything else to do with the SAT, send us an email at info@cardinalec.com .

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