Question

The total surface area of a hemisphere is $108\ \pi\text{cm}^2$. Then what is its volume ?

• A. $288\ \pi\text{cm}^2$
• B. $144\ \pi\text{cm}^2$
• C. $108\ \pi\text{cm}^2$
• D. $72\ \pi\text{cm}^2$

Answer 1

Answer:

$144\ \pi\text{cm}^2$

To find the volume of the hemisphere, we first need to determine its radius ($r$) using the given total surface area.

Step 1: Find the Radius ($r$)

The formula for the total surface area (TSA) of a hemisphere is:
$\text{TSA} = 3\pi r^2$

Given that the TSA is $108\pi\text{ cm}^2$, we set up the equation:
$3\pi r^2 = 108\pi$

Divide both sides by $\pi$:
$3r^2 = 108$

Divide by 3:
$r^2 = \frac{108}{3}$
$r^2 = 36$

Take the square root of both sides:
$r = \sqrt{36} = 6\text{ cm}$

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Step 2: Calculate the Volume

The formula for the volume of a hemisphere is:
$\text{Volume} = \frac{2}{3}\pi r^3$

Substitute the value of $r = 6\text{ cm}$ into the formula:
$\text{Volume} = \frac{2}{3} \times \pi \times (6)^3$
$\text{Volume} = \frac{2}{3} \times \pi \times 216$

Simplify the calculation:
$\text{Volume} = 2 \times \pi \times 72$
$\text{Volume} = \mathbf{144\pi\text{ cm}^3}$

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Final Answer

The volume of the hemisphere is $144\pi\text{ cm}^3$