Question

The area of a semicircle of diameter 'd' is :

• A. d^216
• B. d^24
• C. d^28
• D. d^22

Hint:

Memorizing common shapes' areas in terms of diameter instead of just radius can speed up your calculations.
Area of circle in terms of diameter: \(\frac{\pi d^2}{4}\).
Area of semicircle in terms of diameter: \(\frac{\pi d^2}{8}\).
Area of quadrant in terms of diameter: \(\frac{\pi d^2}{16}\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem asks for the formula representing the area of a semicircle when expressed in terms of its diameter \(d\).

Step 2: Key Formula or Approach:
1. The area of a full circle of radius \(r\) is given by:
\[ A_{\text{circle}} = \pi r^2 \]
2. A semicircle is exactly half of a full circle. Therefore, its area is:
\[ A_{\text{semicircle}} = \frac{1}{2}\pi r^2 \]
3. The relationship between radius \(r\) and diameter \(d\) is:
\[ r = \frac{d}{2} \]
Substituting this relationship into the area equation will yield the required formula.

Step 3: Detailed Explanation:
1. Write down the area of a semicircle as a function of its radius \(r\):
\[ A = \frac{1}{2}\pi r^2 \]
2. Express the radius \(r\) in terms of the diameter \(d\):
\[ r = \frac{d}{2} \]
3. Substitute this expression into the area formula:
\[ A = \frac{1}{2}\pi \left(\frac{d}{2}\right)^2 \]
4. Expand the squared term:
\[ \left(\frac{d}{2}\right)^2 = \frac{d^2}{4} \]
5. Multiply the fractions:
\[ A = \frac{1}{2}\pi \left(\frac{d^2}{4}\right) = \frac{\pi d^2}{8} \]
6. Thus, the area of a semicircle of diameter \(d\) is \(\frac{\pi d^2}{8}\).

Step 4: Final Answer:
Therefore, the correct option is (C).