Question

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

• A. \(\frac{\pi d^2}{16}\)
• B. \(\frac{\pi d^2}{4}\)
• C. \(\frac{\pi d^2}{8}\)
• D. \(\frac{\pi d^2}{2}\)

Hint:

Area of a full circle in terms of diameter is \( \frac{\pi d^2}{4} \). For a semicircle, just multiply by \( \frac{1}{2} \).

Answer 1

The Correct Option is C

To find the area of a semicircle when the diameter is given as \(d\), let's use the concept of a circle's area and apply it to a semicircle.

1. The formula for the area of a circle with diameter \(d\) is: \(\text{Area of circle} = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}\)
2. A semicircle is half of a full circle. Therefore, the area of a semicircle is half the area of the circle.
   • Area of semicircle = \( \frac{1}{2} \times \text{Area of circle}\)
   • Therefore, \(\text{Area of semicircle} = \frac{1}{2} \times \frac{\pi d^2}{4} = \frac{\pi d^2}{8}\)
3. Thus, the area of a semicircle with diameter \(d\) is \(\frac{\pi d^2}{8}\).
4. Among the options provided:
   • \(\frac{\pi d^2}{16}\)
   • \(\frac{\pi d^2}{4}\)
   • \(\frac{\pi d^2}{8}\)
   • \(\frac{\pi d^2}{2}\)
   The correct answer is \(\frac{\pi d^2}{8}\).

Hence, the area of the semicircle with diameter \(d\) is \(\frac{\pi d^2}{8}\).

Remember, for any circle-related problems involving diameter and radius, use the basic area formulae and apply the concept of fractions when dealing with parts of the circle.