Question

the figure shown is formed by three overlapping congruent circles. • the area of the entire figure is 65π square units. • the area of each shaded part is 5π square units. what is the exact area, in square units, of one circle?

Explanation:

Step1: Define variables for circle area

Let the area of one circle be \( A \). Since there are three congruent circles, the total area without considering overlaps is \( 3A \).

Step2: Account for overlapping areas

There are two overlapping (shaded) regions, each with area \( 5\pi \). When we add the three circles, the overlapping parts are counted twice, so we need to subtract the extra count. The total area of the figure is the sum of the three circles minus the overlapping areas (because we added the overlaps twice, we subtract them once). So the equation is \( 3A - 2\times5\pi= 65\pi \) (we subtract \( 2\times5\pi \) because there are two overlapping regions, each counted an extra time).

Step3: Solve the equation for \( A \)

First, simplify the left - hand side of the equation: \( 3A-10\pi = 65\pi \).
Then, add \( 10\pi \) to both sides of the equation: \( 3A=65\pi + 10\pi \).
\( 3A = 75\pi \).
Divide both sides by 3: \( A=\frac{75\pi}{3}=25\pi \).

Answer:

\( 25\pi \)