Express answer in exact form.
Find the area of the larger segment whose chord is 8" long in a circle with an 8" radius. (Hint: A chord divides a circle into two segments. In problem 1, you found the area of the smaller segment.)
I've no idea how to do this, please help me out if you could!
To find the area of the larger segment, we can subtract the area of the smaller segment from the area of the entire circle.
1. Find the area of the smaller segment:
Since the chord is 8" long, it divides the circle into two segments. Let's call the smaller segment "A" and the larger segment "B".
We know that the radius of the circle is 8". Using the formula for the area of a segment, we have:
Area of segment A = (θ - sin(θ)) * r^2 / 2
To find the angle (θ) for the smaller segment, we can use the right triangle formed by the radius, chord, and the perpendicular bisector of the chord. The angle θ is equal to twice the arcsine of half the length of the chord divided by the radius.
θ = 2 * arcsin(chord/2*radius)
θ = 2 * arcsin(8/2*8)
= 2 * arcsin(1/2)
= 2 * arcsin(0.5)
≈ 1.047 radians
The area of segment A = (1.047 - sin(1.047)) * 8^2 / 2
= (1.047 - 0.867) * 64 / 2
= 0.18 * 64 / 2
= 11.52 square inches
2. Find the area of the entire circle:
The area of a circle is given by the formula A = π * r^2.
Area of the entire circle = π * 8^2
= π * 64
≈ 201.06 square inches
3. Find the area of the larger segment:
Area of segment B = Area of the entire circle - Area of segment A
= 201.06 - 11.52
≈ 189.54 square inches
So, the area of the larger segment, when the chord is 8" long in a circle with an 8" radius, is approximately 189.54 square inches.
To find the area of a segment of a circle, you can use the formula:
Area = (θ/360) * π * r² - (1/2) * r² * sin(θ)
where θ is the central angle in degrees, r is the radius of the circle, and π is a mathematical constant approximately equal to 3.14159.
In this case, the chord is 8" long, and the radius of the circle is also 8". Since the chord is equal to the diameter in length, we know that it subtends a 180° angle at the center of the circle.
Now, we can use the formula to calculate the area of the larger segment:
Area = (180/360) * π * (8)² - (1/2) * (8)² * sin(180)
Simplifying this expression step by step:
Area = (1/2) * π * (8)² - (1/2) * (8)² * 1
Area = (1/2) * π * 64 - (1/2) * 64
Area = 32π - 32
So, the area of the larger segment is given by 32π - 32 square inches in exact form.
Given a circle with an 8" radius, find the area of the smaller segment whose chord is 8" long
If you did problem 1, then just subtract that from the area of the whole circle.
If not, let's do it now. Recall that the area of the smaller segment is
a = 1/2 r^2 (θ-sinθ)
So, what's θ? Draw your diagram, and you can easily see that
sin(θ/2) = 4/8 = 1/2
θ = π/3
Plug that into the formula, and you will have the area of the smaller segment.