A semicircle is constructed along each side of a right triangle with legs 6 inches and 8 inches. The semicircle placed along the hypotenuse is shaded, as shown. What is the total area of the two non-shaded crescent-shaped regions? Express your answer in simplest form.
Area of each semicircle = 1/2 π r^2
r = 1/2 diameter
I'll let you do the calculations.
To find the total area of the two non-shaded crescent-shaped regions, we first need to find the area of the semicircle placed along the hypotenuse.
The formula for the area of a semicircle is (π * r^2) / 2, where r is the radius of the semicircle.
In this case, the hypotenuse of the right triangle is equal to the diameter of the semicircle. We can find the hypotenuse using the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the lengths of the legs of the triangle, and c is the length of the hypotenuse.
For this right triangle, the lengths of the legs are 6 inches and 8 inches:
c^2 = 6^2 + 8^2
c^2 = 36 + 64
c^2 = 100
c = √100
c = 10 inches
So, the diameter of the semicircle is 10 inches, and the radius is half the diameter, which is 5 inches.
Now, let's calculate the area of the shaded semicircle:
Area of shaded semicircle = (π * r^2) / 2
Area of shaded semicircle = (π * 5^2) / 2
Area of shaded semicircle = (π * 25) / 2
Area of shaded semicircle = 12.5π square inches
To find the total area of the two non-shaded crescent-shaped regions, we need to subtract the area of the shaded semicircle from the total area of the two semicircles.
The total area of the two semicircles is given by the formula (π * r^2) / 2 + (π * r^2) / 2 = π * r^2, since the radius of each semicircle is the same.
Total area of two semicircles = 2 * (π * 5^2)
Total area of two semicircles = 2 * (π * 25)
Total area of two semicircles = 50π square inches
Now, subtract the area of the shaded semicircle from the total area of the two semicircles:
Total area of the two non-shaded crescent-shaped regions = Total area of two semicircles - Area of shaded semicircle
Total area of the two non-shaded crescent-shaped regions = 50π - 12.5π
Total area of the two non-shaded crescent-shaped regions = 37.5π square inches
Therefore, the total area of the two non-shaded crescent-shaped regions is 37.5π square inches.