Find Perimeter of shaded region in semicircle.

Question

What is the Perimeter of shaded region in semicircle if four small semicircles have radii of 1,2,3,4 respectively?

a. 10 $\pi$ b. 20 $\pi$ c. 40 $\pi$ d. 60 $\pi$

Answer 1

Add all radii and then multiply by $\pi$:

$$\pi[1+2+3+4+(1+2+3+4)]$$

Answer 2

It`s A because (1 + 2 + 3 + 4) = 10 hence 10pi is perimeter.

Answer 3

Since the perimeter of a half-circle is just $\pi$ times the diameter, the perimeter of the shaded region equals the perimeter of the whole big circle. The big circle has radius $10$, hence the answer is $\color{red}{20\pi}$.

Answer 4

perimeter of circle = 2πr (or πd, where d-> diameter)

perimeter of shaded region = (sum of perimeter of all circles) / 2

how many circles are there ?

a. circle with radius 1  (perimeter = 2πr = 2π)
b. circle with radius 2  (perimeter = 2πr = 4π)
c. circle with radius 3  (perimeter = 2πr = 6π)
d. circle with radius 4  (perimeter = 2πr = 8π)
e. circle with radius (1 + 2 + 3 + 4)  (perimeter = 2πr = 20π)

perimeter of shaded region

=  (sum of perimeter of all circles) / 2
=  (a + b + c + d) / 2
=  (2π + 4π + 6π + 8π + 20π) / 2
=  20π