Find the area of the shaded region. Be sure to write the formula, work and answer with
units. the arc length is 135 degrees and the diameter of the circle is 18cm and half of the circle is not the shaded region
and the 135 degrees arc isn't the shaded region either.
The area of the shaded region can be found by first calculating the area of the entire circle and then subtracting the area of the unshaded region (which includes half of the circle and the 135 degrees arc).
1. The formula to calculate the area of a circle is:
Area = πr^2
where r is the radius of the circle.
2. Since the diameter of the circle is 18 cm, the radius (r) is half of the diameter:
r = 18 cm / 2 = 9 cm
3. Calculate the area of the entire circle:
Area = π(9 cm)^2
Area = π(81 cm^2)
Area ≈ 254.47 cm^2
4. Now, we need to calculate the area of the unshaded region.
The area of half of the circle is:
Area of half circle = 0.5 * Area of the entire circle
Area of half circle = 0.5 * 254.47 cm^2
Area of half circle ≈ 127.24 cm^2
5. The area of the 135 degrees arc can be calculated using the formula:
Arc Length = (θ/360) * 2πr
where θ is the central angle in radians.
6. Convert 135 degrees to radians:
θ = 135 degrees * (π/180)
θ ≈ 2.36 radians
7. Calculate the arc length of the 135 degrees arc:
Arc Length = (2.36 / 2π) * 2π * 9 cm
Arc Length ≈ 13.43 cm^2
8. Calculate the total unshaded area:
Total unshaded area = Area of half circle + Arc Length
Total unshaded area ≈ 127.24 cm^2 + 13.43 cm^2
Total unshaded area ≈ 140.67 cm^2
9. Finally, find the area of the shaded region by subtracting the total unshaded area from the area of the entire circle:
Area of shaded region = Area of the entire circle - Total unshaded area
Area of shaded region ≈ 254.47 cm^2 - 140.67 cm^2
Area of shaded region ≈ 113.80 cm^2
Therefore, the area of the shaded region is approximately 113.80 cm^2.
make less work
Sure! Here is a more streamlined approach to find the area of the shaded region:
1. Calculate the area of the entire circle:
Area = π(9 cm)^2
Area = 81π cm^2 ≈ 254.47 cm^2
2. Calculate the area of the unshaded region (half of the circle + 135 degrees arc):
Area of unshaded region = 0.5 * 81π cm^2 + (135/360) * π(9 cm)^2
Area of unshaded region ≈ 127.24 cm^2 + 3.98 cm^2
Area of unshaded region ≈ 131.22 cm^2
3. Calculate the area of the shaded region by subtracting the unshaded area from the entire circle:
Area of shaded region = 254.47 cm^2 - 131.22 cm^2
Area of shaded region ≈ 123.25 cm^2
Therefore, the area of the shaded region is approximately 123.25 cm^2.