Question

The total surface area of a solid hemisphere of diameter ‘2d’ is :

• A. \(3\pi d^2\)
• B. \(2\pi d^2\)
• C. \(\frac{1}{2}\pi d^2\)
• D. \(\frac{3}{4}\pi d^2\)

Hint:

Don't confuse Total Surface Area (\(3\pi r^2\)) with Curved Surface Area (\(2\pi r^2\)). For a solid hemisphere, the base area \(\pi r^2\) must be included.

Answer 1

The Correct Option is A

To find the total surface area of a solid hemisphere with a given diameter, we start by understanding the geometry of the hemisphere. 

The formula for the total surface area of a solid hemisphere is obtained by adding the curved surface area and the area of the circular base.

• The curved surface area (CSA) of a hemisphere is given by the formula: \(\text{CSA} = 2\pi r^2\), where \(r\) is the radius of the hemisphere.
• The area of the base is nothing but the area of a circle: \(\text{Area of base} = \pi r^2\).

First, we need to compute the total surface area by adding these two areas:

• \(\text{Total Surface Area} = \text{CSA} + \text{Area of base} = 2\pi r^2 + \pi r^2 = 3\pi r^2\)

Given that the diameter of the solid hemisphere is \(2d\), the radius \(r\) becomes:

• \(r = \frac{2d}{2} = d\)

Substitute \(r = d\) into the total surface area formula:

• \(\text{Total Surface Area} = 3\pi d^2\)

Thus, the total surface area of the solid hemisphere with diameter \(2d\) is \(3\pi d^2\).

Therefore, the correct answer is

\(3\pi d^2\)

.