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:

Be extremely careful to differentiate between "solid hemisphere" and "hollow hemisphere".
For a hollow hemisphere, the surface area is just the curved surface area, $2\pi r^2$.
For a solid hemisphere, always include the area of the flat base ($\pi r^2$), which brings the total to $3\pi r^2$.
Also, always make sure to convert the given diameter into radius before applying the standard formulas.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The topic is Surface Areas and Volumes.
We are asked to find the formula for the Total Surface Area (TSA) of a solid hemisphere when its diameter is given in terms of the variable $'2d'$.

Step 2: Key Formula or Approach:
A solid hemisphere has two surfaces:

• A curved surface with area equal to $2\pi r^2$.

• A flat circular base with area equal to $\pi r^2$.

The Total Surface Area (TSA) of a solid hemisphere is the sum of these two areas:
\[ \text{TSA} = 2\pi r^2 + \pi r^2 = 3\pi r^2 \]
where $r$ is the radius of the hemisphere.
We must relate the radius $r$ to the given diameter $D = 2d$ and substitute it into the formula.

Step 3: Detailed Explanation:

• Let the diameter of the solid hemisphere be $D$. We are given:
\[ D = 2d \]

• The relationship between the radius $r$ and the diameter $D$ of a sphere/hemisphere is:
\[ r = \frac{D}{2} \]

• Substitute the value of $D = 2d$ into the relation:
\[ r = \frac{2d}{2} = d \]

• Write down the standard formula for the Total Surface Area of a solid hemisphere:
\[ \text{TSA} = 3\pi r^2 \]

• Substitute the value of $r = d$ into the surface area equation:
\[ \text{TSA} = 3\pi (d)^2 \]
\[ \text{TSA} = 3\pi d^2 \]


Step 4: Final Answer:
The total surface area of the solid hemisphere is $3\pi d^2$, which corresponds to option (A).