In circle TT, the length of arc, U, V, equals, start fraction, 3, divided by, 4, end fraction, pi
UV
⌢
=
4
3
π and mangle, U, T, V, equals, 45, degrees∠UTV=45
∘
. Find the area shaded below. Express your answer as a fraction times piπ.
thats not correct
I apologize for the mistake in the previous calculation. Let's re-calculate the shaded area correctly.
Given:
- Length of arc UV = (3/4)π
- Central angle ∠UTV = 45 degrees
1. Radius calculation:
Since the measure of arc UV is (3/4)π and the central angle ∠UTV is 45 degrees, the circumference of the whole circle would be (4/3) * (180/45) = 16 units.
Hence, the radius of the circle would be 16 / (2π) = 8 / π units.
2. Area of sector UTV:
The area of the sector UTV can be calculated by using the formula A = (θ/360)πr^2.
Substituting the values, we get:
A_sector = (45/360) * π * (8/π)^2
A_sector = (1/8) * 64
A_sector = 8 square units.
3. Area of triangle UTV:
Since the central angle ∠UTV is 45 degrees, UTV is an isosceles triangle.
The area of an isosceles triangle can be calculated by: A = 0.5 * a * b * sin(C), where C is the angle between sides a and b.
Here, the sides a and b are radii, and the angle between them is 45 degrees.
Substituting the values, we get:
A_triangle = 0.5 * (8/π) * (8/π) * sin(45 degrees)
A_triangle = 0.5 * (64 / π^2) * sqrt(2) / 2
A_triangle = 32sqrt(2) / (2π^2) = 16sqrt(2) / π^2 square units.
4. Shaded area calculation:
The shaded area is the area of sector UTV minus the area of triangle UTV:
Shaded area = A_sector - A_triangle
Shaded area = 8 - 16sqrt(2) / π^2
Shaded area = (8π^2 - 16sqrt(2)) / π^2 square units.
Therefore, the correct shaded area below is (8π^2 - 16sqrt(2)) / π^2 square units.
To find the area shaded below, we need to find the area of sector UTV and subtract the area of triangle UTV.
1. Area of sector UTV:
The area of a sector of a circle is given by the formula A = (θ/360)πr^2, where θ is the central angle in degrees and r is the radius of the circle.
Given that the length of arc UV is (3/4)π and the central angle is 45 degrees, the radius can be calculated as follows:
(3/4)π = r * (45/360) * π
r = (3/4) * (45/360)
r = 3/16
Now, we can find the area of sector UTV:
A_sector = (45/360) * π * (3/16)^2
A_sector = (1/8)π
2. Area of triangle UTV:
To find the area of the triangle UTV, we can use the formula for the area of a triangle: A = 0.5 * base * height.
The height of the triangle is the radius of the circle, which we found to be 3/16. The base can be calculated using trigonometry:
tan(45 degrees) = (height of triangle) / (base of triangle)
1 = (3/16) / base
base = 3/16
Now we can find the area of the triangle:
A_triangle = 0.5 * (3/16) * (3/16)
A_triangle = 9/256
3. Subtract the area of triangle from the area of the sector to find the shaded area:
Shaded area = A_sector - A_triangle
Shaded area = (1/8)π - 9/256
Shaded area = (32/256)π - 9/256
Shaded area = (23/256)π
Therefore, the area shaded below is (23/256)π.