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Find the curved surface area of a cone whose base radius is 7 cm and slant height is 10 cm. (Use pi = 22/7)

220 $\text{ cm}^2$

440 $\text{ cm}^2$

154 $\text{ cm}^2$

308 $\text{ cm}^2$

Correct Answer

📐 MA • Math Concept & PrincipleGeneral ScienceMathematics

Option A

220 $\text{ cm}^2$

Quick Summary:

The curved surface area (CSA) of a cone is defined as the area of the lateral surface excluding the circular base. Using the radius $r = 7 \text{ cm}$ and slant height $l = 10 \text{ cm}$ , the CSA is calculated as $\pi rl = \frac{22}{7} \times 7 \times 10 = 220 \text{ cm}^2$ .

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🔬SCScience SolutionConcept & Principle

💡 Explanation

💡 Everyday Analogy (Real-World Intuition)

Think of a birthday party hat; the paper needed to make the cone without including the bottom base is the curved surface area.

🧠 Memory Mnemonic / Shortcut Aid

Remember 'Pie-RL' (πrl) for the curved surface to keep it 'real'.

🔢 Key Formulas

$CSA = \pi rl$ — Curved Surface Area of a cone

$TSA = \pi r(r + l)$ — Total Surface Area including the circular base

$l = \sqrt{r^2 + h^2}$ — Slant height relation using Pythagorean theorem

⚙️ Working Principle

The surface area of a cone is derived from the net of a circular sector. When a sector of a circle with radius equal to the cone's slant height $l$ is wrapped, the arc length corresponds to the circumference of the cone's base $2\pi r$ . Integrating this geometry leads to the formula $CSA = \pi rl$ .

📌 Key Points

▸
Slant height $l$ is the distance from the apex to any point on the circumference of the base.
▸
The constant $\pi$ is approximately $\frac{22}{7}$ for calculation purposes in competitive exams.
▸
Area is always measured in square units (e.g., $\text{cm}^2$ ).

✅ Advantages

▸
Formula allows rapid calculation of material requirements for conical structures.

❌ Disadvantages / Limitations

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Only applies to right circular cones where slant height is uniform.

🛠️ Applications / Uses

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Calculation of fabric used in tents.
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Designing conical funnels and traffic cones.

📄 Additional Information

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Standard value of $\pi \approx 3.14159$ .
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Option B (440) is $2 \times \pi rl$ , which is incorrect. Option C (154) is the area of the base ( $\pi r^2$ ). Option D (308) is an incorrect derivation.

📊 Diagram / Illustration

✅

A is correct — The curved surface area is calculated by multiplying $\pi$ , radius, and slant height, resulting in $220 \text{ cm}^2$ .

Core Concepts Used

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Mensuration Geometry of solids Surface area derivation

💡 EXAM TIP

Always verify if the question asks for 'Curved Surface Area' (only the side) or 'Total Surface Area' (side + base) to avoid common calculation traps.

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