Question

Express answer in exact form. A segment of a circle has a 120 arc and a chord of 8in. Find the area of the segment.

Answer 1

Answer:

13.09 in²

Step-by-step explanation:

Formula to find the Area of Segment:

• A= (θ × π/360° − sin(θ)/2) × r², where θ = arc in degrees, r = radius of circle

• θ = 120°

We need to find the radius of the circle for our calculation of area of the segment.

Refer to picture.

The right triangle ΔOBC has:

• BC= 1/2 of chord AB = 8 in/2 = 4 in
• ∠COB = 1/2 θ= 1/2*120° = 60°

Since ΔOBC is 30-60-90 triangle:

• 4 = √3/2r
• r = 8√3/3 in

Then the area of segment:

• A= (θ × π/360° − sin(θ)/2) × r² =
• (120° × π/360° − sin(120°)/2) × (8√3/3)² =
• (3.14/3 - √3/4)× 64/3 =
• 13.09 in²

Answer: 13.09 in²