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: Treat the semicircle as a sector with a 180 degree angle.
A semicircle is a special case of a circular sector whose central angle is $180^\circ$, half of a full revolution. We can reach the formula through the general sector-area formula instead of halving the whole circle's area directly.

Step 2: Write the general sector-area formula.
\[ \text{Area} = \frac{\theta}{360^\circ}\times\pi r^2 \]

Step 3: Substitute theta = 180 degrees.
\[ \text{Area of semicircle} = \frac{180^\circ}{360^\circ}\times\pi r^2 = \frac12\pi r^2 \]

Step 4: Express the radius in terms of the diameter.
Since $r=\dfrac{d}{2}$:
\[ \text{Area} = \frac12\pi\left(\frac{d}{2}\right)^2 = \frac12\pi\cdot\frac{d^2}{4} = \frac{\pi d^2}{8} \]

Final Answer:
The area of the semicircle is $\dfrac{\pi d^2}{8}$, matching option (C).
\[ \boxed{\frac{\pi d^2}{8}} \]