Question

two small circles lie on the diameter of a larger circle as shown. if ab = 24 cm, what is the approximate area of the shaded region?
options:
282.74 cm²
169.65 cm²
113.10 cm²
226.19 cm²

Explanation:

Step1: Find radius of large circle

The diameter of the large circle \( AB = 24 \, \text{cm} \), so radius \( R=\frac{24}{2}=12 \, \text{cm} \).

Step2: Find radius of small circles

Each small circle has diameter \( \frac{24}{2}=12 \, \text{cm} \), so radius \( r = \frac{12}{2}=6 \, \text{cm} \).

Step3: Area of large circle

Using formula \( A_{\text{large}}=\pi R^{2} \), substitute \( R = 12 \):
\( A_{\text{large}}=\pi(12)^{2}=144\pi \, \text{cm}^2 \).

Step4: Area of two small circles

Each small circle area \( A_{\text{small}}=\pi r^{2}=\pi(6)^{2}=36\pi \), so two small circles:
\( A_{\text{two small}} = 2\times36\pi = 72\pi \, \text{cm}^2 \).

Step5: Area of shaded region

Shaded area \( = A_{\text{large}} - A_{\text{two small}} = 144\pi - 72\pi = 72\pi \).
Approximate \( \pi\approx3.1416 \), so \( 72\times3.1416\approx226.19 \, \text{cm}^2 \).

Answer:

\( 226.19 \, \text{cm}^2 \) (the option: \( 226.19 \, \text{cm}^2 \))