A circle with a radius of 8 centimeters. Two radii form a right angle. The area between them is shaded.
Question
What is the area of the shaded region, rounded to the nearest tenth?
The area of the shaded region can be found by subtracting the area of the quarter circle from the area of the right triangle formed by the two radii.
The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
The area of a quarter circle with a radius of 8 centimeters is therefore Aq = (1/4) * π * 8^2 = 16π cm^2.
The area of a right triangle is given by the formula A = (1/2) * base * height.
The base of the right triangle is one of the radii, which has a length of 8 centimeters.
The height of the right triangle can be found using the Pythagorean theorem. The other leg of the triangle is also a radius with a length of 8 centimeters, so the hypotenuse of the right triangle is the diameter of the circle, which has a length of 16 centimeters.
Using the Pythagorean theorem, we can find the height of the right triangle:
8^2 + h^2 = 16^2
64 + h^2 = 256
h^2 = 256 - 64
h^2 = 192
h = √192
h ≈ 13.9 centimeters
The area of the right triangle is therefore Atr = (1/2) * 8 * 13.9 = 55.6 cm^2.
The area of the shaded region is the difference between the area of the quarter circle and the area of the right triangle:
Ashaded = Aq - Atr
Ashaded = 16π - 55.6
Rounding to the nearest tenth, the area of the shaded region is approximately 16π - 55.6 ≈ 146.7 cm^2.