The area of a semicircle is half the area of a circle. Note that the area of the unshaded region is equal to the area of the larger semicircle minus the area of the triangle. You will need to use the Pythagorean Theorem.
See solution.
Practice makes perfect
Let a, b, and c be the radii of each semicircle. Then, the diameters of the semicircles are 2a, 2b, and 2c, respectively. Let's label these radii and the shaded areas.
Since we have a right triangle, we can apply the Pythagorean Theorem and write the following equation.
(2c)^2 = (2a)^2 + (2b)^2
⇓
c^2 = a^2 + b^2The area of the right triangle is equal to one-half the product of the length of the legs.
A_3 = 1/2* 2a* 2b
⇓
A_3 = 2ab
Now, let's find the area of the three semicircles. Keep in mind that their areas are half the area of the corresponding circle.
Radius
Area of Semicirle
a
A_(S_1) = 1/2Ï€ a^2
b
A_(S_2) = 1/2Ï€ b^2
c
A_(S_3) = 1/2Ï€ c^2
From the diagram, we can see that the area of the unshaded region is equal to the area of the semicircle with radius c minus the area of the triangle.
A_(unshaded) = A_(S_3) - A_3
Let's substitute the corresponding values into the equation above and simplify it.
Finally, to find the area of the two shaded crescents we subtract the unshaded area from the sum of the areas of the two smaller semicircles.
A_1 + A_2_(crescents) = A_(S_1) + A_(S_2) - A_(unshaded)
As before, let's substitute and simplify the corresponding values into the equation above.
