A segment of a circle has a 120 arc and a chord of 8in. Find the area of the segment
Did you mean the arc has a central angle of 120° ?
I cannot draw a segment with arclength 120 and chord of 8
To find the area of a segment, we can use the following formula:
Area = (θ/360) × πr² - 0.5r²sin(θ)
Where:
- θ is the measure of the arc in degrees
- r is the radius of the circle
First, we need to find the radius of the circle. Since we are given the length of a chord, we can use the following formula to find the radius:
r = (c/2sin(θ/2))
Where:
- c is the length of the chord
- θ is the measure of the arc in degrees
Given that the chord has a length of 8 inches and the arc is 120 degrees, we can find the radius:
r = (8/2sin(120/2))
Now, we can calculate the value of r:
r = (8/2sin(60))
r = (8/2 × (√3) / 2)
r = (8/2) × (√3/2)
r = 4 × (√3/2)
r = 2√3
Now that we have the radius, we can substitute the values into the area formula:
Area = (θ/360) × πr² - 0.5r²sin(θ)
Area = (120/360) × π(2√3)² - 0.5(2√3)²sin(120)
Simplifying further:
Area = (1/3) × 4π - 0.5(12) × (√3/2)
Area = (4/3)π - 6√3
Therefore, the area of the segment of the circle is (4/3)π - 6√3 square units.