Answer has to be in exact form.Find the area of a segment formed by a chord 8" long in a circle with radius of 8".
look at the triangle formed by the chord and the radii.
Don't you have an equilateral triangle ?
That would make the central angle = 60°
So the area of the SECTOR is 1/6 of the area of the circle.
If you subtract the area of the equilateral triangle you would be left with the area of the segment
In your previous post, I showed you how to find the height of such a triangle.
So the problem should be easy for you now.
To find the area of a segment formed by a chord in a circle, we need to use the formula:
Area of segment = (θ - sin(θ)) * (r^2) / 2
Where:
- θ is the central angle in radians
- r is the radius of the circle
We are given that the chord is 8 inches long and the radius is 8 inches. Let's use these values to find θ.
First, we need to find the central angle using the formula:
θ = 2 * arcsin(chord / (2 * r))
Plugging in the values:
θ = 2 * arcsin(8 / (2 * 8))
θ = 2 * arcsin(1)
θ = 2 * (pi/2)
θ = pi
Now we can substitute θ and r into the area formula:
Area of segment = (π - sin(π)) * (8^2) / 2
Area of segment = (π - 0) * (64) / 2
Area of segment = π * 32
Thus, the area of the segment formed by the chord 8 inches long in a circle with a radius of 8 inches is approximately 100.53 square inches.