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If the radius of a sphere is doubled, what happens to its surface area?

Doubles

Triples

Quadruples

Remains same

Correct Answer

📐 MA • Math Concept & PrincipleGeneral ScienceMathematics

Option C

Quadruples

Quick Summary:

The surface area of a sphere is proportional to the square of its radius ( $r^2$ ). When the radius is doubled ( $2r$ ), the new surface area becomes $4$ times the original value.

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🔬SCScience SolutionConcept & Principle

💡 Explanation

The surface area of a sphere is proportional to the square of its radius ( $r^2$ ). When the radius is doubled ( $2r$ ), the new surface area becomes $4$ times the original value.

💡 Everyday Analogy (Real-World Intuition)

If you double the width and height of a square piece of paper, you get 4 pieces of paper that fit into the area of the new shape.

🧠 Memory Mnemonic / Shortcut Aid

Square the change: if radius is scaled by factor $k$ , Area is scaled by $k^2$ and Volume is scaled by $k^3$ .

🔢 Key Formulas

$A = 4\pi r^2$ — Surface area of a sphere

$V = \frac{4}{3}\pi r^3$ — Volume of a sphere

⚙️ Working Principle

The surface area $A$ of a sphere is calculated using the formula $A = 4\pi r^2$ . Substituting the new radius $r' = 2r$ into the formula yields $A' = 4\pi(2r)^2 = 4\pi(4r^2) = 16\pi r^2$ . Since the original area is $4\pi r^2$ , the new area $A'$ is $4 \times A$ .

📌 Key Points

▸
Surface area scales with the square of the linear dimension.
▸
Volume scales with the cube of the linear dimension.
▸
The constant $4\pi$ remains unchanged during scaling.

✅ Advantages

▸
Easy calculation for geometric scaling problems.

❌ Disadvantages / Limitations

▸
Does not apply to non-spherical objects with different scaling ratios.

🛠️ Applications / Uses

▸
Heat dissipation efficiency in spherical tanks.
▸
Drug delivery particles where surface-to-volume ratio dictates release rates.

📄 Additional Information

▸
If the radius is tripled, the area increases by a factor of $3^2 = 9$ .
▸
Option A is incorrect because it ignores the square dependency.
▸
Option B is incorrect because it fails to calculate the geometric power of 2.

📊 Diagram / Illustration

✅

C is correct — The surface area of a sphere follows the square law, so doubling the radius results in a fourfold increase in area.

Core Concepts Used

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Geometric Scaling Surface Area Formula Square Law

💡 EXAM TIP

For competitive exams, remember: Area scales by $k^2$ and Volume scales by $k^3$ when linear dimensions are scaled by factor $k$ .

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