The total surface area of a solid hemisphere is 462 cm^2. Find its radius. (Use pi = 22/7)

The correct answer is Option A: 7 cm. 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. Given the surface area $462 \text{ cm}^2$ and using $\pi = \frac{22}{7}$, the radius $r$ is calculated to be $7 \text{ cm}$ by solving the equation $3\pi r^2 = 462$. Step-by-Step Solution & Concepts: 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. Given the surface area $462 \text{ cm}^2$ and using $\pi = \frac{22}{7}$, the radius $r$ is calculated to be $7 \text{ cm}$ by solving the equation $3\pi r^2 = 462$. Working Principle: A hemisphere is exactly half of a sphere. While the curved surface area is $2\pi r^2$, a solid hemisphere includes the flat circular top of area $\pi r^2$, resulting in a total surface area formula of $3\pi r^2$. By substituting the given values, we get $3 \times \frac{22}{7} \times r^2 = 462$, which simplifies to $r^2 = 49$, leading to $r = 7$. Key Formulas: $TSA = 3\pi r^2$ — Total surface area of a solid hemisphere; $CSA = 2\pi r^2$ — Curved surface area of a hemisphere Key Points: A solid hemisphere includes the base area, whereas a hollow hemisphere does not.; The radius is the distance from the center of the circular base to any point on the curved surface.; The surface area is measured in square units while the radius is measured in linear units. Advantages: Simplifies calculation of complex 3D shapes by breaking them into standard geometric components. Disadvantages: Requires knowledge of constant $\pi$ and exact geometric formulas. Applications: Structural engineering for domes and roof designs.; Manufacturing of hemispherical containers or kitchenware. Additional Info: The value $\pi \approx 3.14$ or $\frac{22}{7}$ is a standard mathematical constant used for circular geometry.. Option B (14 cm) results in $3 \times \frac{22}{7} \times 196 = 1848$, which is incorrect.. Option C (21 cm) results in $3 \times \frac{22}{7} \times 441 = 4158$, which is incorrect.. Option D (3.5 cm) results in $3 \times \frac{22}{7} \times 12.25 = 115.5$, which is incorrect. Answer: A is correct — The calculation $3 \times \frac{22}{7} \times r^2 = 462$ yields $r^2 = 49$, confirming the radius is $7 \text{ cm}$. Exam Tip: Always verify if the question specifies a 'solid' hemisphere or a 'hollow/curved' surface area, as this changes the formula from $3\pi r^2$ to $2\pi r^2$.

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The total surface area of a solid hemisphere is 462 cm². Find its radius. (Use $\pi$ = 22/7)

7 cm

14 cm

21 cm

3.5 cm

Correct Answer

📐 MA • Math Concept & PrincipleGeneral ScienceMathematics

Option A

7 cm

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. Given the surface area $462 \text{ cm}^2$ and using $\pi = \frac{22}{7}$ , the radius $r$ is calculated to be $7 \text{ cm}$ by solving the equation $3\pi r^2 = 462$ .

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🔬SCScience SolutionConcept & Principle

💡 Explanation

💡 Everyday Analogy (Real-World Intuition)

Think of a solid hemisphere like an orange cut exactly in half: the 'skin' covers the round part, but the flat cut surface also requires covering, making the total surface area larger than just the dome part.

🧠 Memory Mnemonic / Shortcut Aid

Remember '3PI R squared' (3 PIR2) for a solid, versus '2PI R squared' for just the bowl-like curved surface.

🔢 Key Formulas

$TSA = 3\pi r^2$ — Total surface area of a solid hemisphere

$CSA = 2\pi r^2$ — Curved surface area of a hemisphere

⚙️ Working Principle

A hemisphere is exactly half of a sphere. While the curved surface area is $2\pi r^2$ , a solid hemisphere includes the flat circular top of area $\pi r^2$ , resulting in a total surface area formula of $3\pi r^2$ . By substituting the given values, we get $3 \times \frac{22}{7} \times r^2 = 462$ , which simplifies to $r^2 = 49$ , leading to $r = 7$ .

📌 Key Points

▸
A solid hemisphere includes the base area, whereas a hollow hemisphere does not.
▸
The radius is the distance from the center of the circular base to any point on the curved surface.
▸
The surface area is measured in square units while the radius is measured in linear units.

✅ Advantages

▸
Simplifies calculation of complex 3D shapes by breaking them into standard geometric components.

❌ Disadvantages / Limitations

▸
Requires knowledge of constant $\pi$ and exact geometric formulas.

🛠️ Applications / Uses

▸
Structural engineering for domes and roof designs.
▸
Manufacturing of hemispherical containers or kitchenware.

📄 Additional Information

▸
The value $\pi \approx 3.14$ or $\frac{22}{7}$ is a standard mathematical constant used for circular geometry.
▸
Option B (14 cm) results in $3 \times \frac{22}{7} \times 196 = 1848$ , which is incorrect.
▸
Option C (21 cm) results in $3 \times \frac{22}{7} \times 441 = 4158$ , which is incorrect.
▸
Option D (3.5 cm) results in $3 \times \frac{22}{7} \times 12.25 = 115.5$ , which is incorrect.

📊 Diagram / Illustration

✅

A is correct — The calculation $3 \times \frac{22}{7} \times r^2 = 462$ yields $r^2 = 49$ , confirming the radius is $7 \text{ cm}$ .

Core Concepts Used

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Mensuration of 3D solids Surface area geometry Algebraic solving of quadratic equations

💡 EXAM TIP

Always verify if the question specifies a 'solid' hemisphere or a 'hollow/curved' surface area, as this changes the formula from $3\pi r^2$ to $2\pi r^2$ .

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