The total surface area of a solid hemisphere is $ 191\text{ cm... | General Science MCQ

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The total surface area of a solid hemisphere is $191\text{ cm}^2$ . Its radius (in cm) is: (Take $\pi = \frac{22}{7}$ )

$\sqrt{\frac{1347}{66}}$

$\sqrt{\frac{1347}{132}}$

$\sqrt{\frac{1337}{66}}$

$\sqrt{\frac{1345}{66}}$

Correct Answer

🔬 SC • Science Concept & PrincipleGeneral SciencePrevious Year Paper 2024

Option C

$\sqrt{\frac{1337}{66}}$

Quick Summary:

The total surface area of a solid hemisphere is given by the sum of its curved surface area and the area of its circular base, which is $3\pi r^2$ . By setting $3\pi r^2 = 191$ and using the value $\pi = \frac{22}{7}$ , we solve for $r^2 = \frac{191 \times 7}{3 \times 22} = \frac{1337}{66}$ .

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🔬SCScience SolutionConcept & Principle

💡 Explanation

💡 Everyday Analogy (Real-World Intuition)

Think of a bowl filled with soup; the 'total surface area' is the outer round shell plus the surface of the soup at the top.

🔢 Key Formulas

$TSA = 3\pi r^2$ — Total Surface Area of a solid hemisphere

$r = \sqrt{\frac{TSA}{3\pi}}$ — Formula to calculate radius from surface area

⚙️ Working Principle

A solid hemisphere consists of a curved surface area equal to $2\pi r^2$ (half the surface area of a sphere) and a flat circular base area of $\pi r^2$ . Summing these results in the formula $TSA = 3\pi r^2$ . To isolate the radius $r$ , we rearrange the equation as $r = \sqrt{\frac{TSA}{3\pi}}$ .

📌 Key Points

▸
A solid hemisphere has a base, unlike a hollow shell.
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The total surface area includes the curved part ( $2\pi r^2$ ) and the base ( $\pi r^2$ ).
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Substituting $\pi = \frac{22}{7}$ simplifies the calculation of the radius variable.

🛠️ Applications / Uses

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Geometry of spherical containers
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Hemispherical structural engineering

📄 Additional Information

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The calculation simplifies to $3 \times \frac{22}{7} \times r^2 = 191$ .
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Option C is the mathematically derived result $\sqrt{\frac{1337}{66}}$ .

📊 Diagram / Illustration

✅

C is correct — The total surface area of a solid hemisphere is $3\pi r^2$ , leading to $r^2 = \frac{1337}{66}$ , so $r = \sqrt{\frac{1337}{66}}$ .

Core Concepts Used

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Solid Hemisphere Surface Area Algebraic Rearrangement

💡 EXAM TIP

Always remember the difference between curved surface area ( $2\pi r^2$ ) and total surface area ( $3\pi r^2$ ) for hemispheres, as this is a common point of error in competitive exams.

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