Surface area and volume of a cone with slant height — derivation and numerical

Question

A cone has radius $r = 7$ cm and slant height $l = 25$ cm. Find:

The curved surface area (CSA)
The total surface area (TSA)
The volume

(NCERT Class 10, Chapter 13 — Surface Areas and Volumes)

Solution — Step by Step

For any right circular cone, the radius $r$ , height $h$ , and slant height $l$ form a right triangle:

l^2 = r^2 + h^2

We need height for the volume, so: $h = \sqrt{l^2 - r^2} = \sqrt{625 - 49} = \sqrt{576} = 24$ cm.

\text{CSA} = \pi r l

\text{CSA} = \frac{22}{7} \times 7 \times 25 = 22 \times 25 = \mathbf{550 \text{ cm}^2}

\text{TSA} = \pi r l + \pi r^2 = \pi r(l + r)

\text{TSA} = \frac{22}{7} \times 7 \times (25 + 7) = 22 \times 32 = \mathbf{704 \text{ cm}^2}

V = \frac{1}{3}\pi r^2 h

V = \frac{1}{3} \times \frac{22}{7} \times 7^2 \times 24 = \frac{1}{3} \times \frac{22}{7} \times 49 \times 24

= \frac{1}{3} \times 22 \times 7 \times 24 = \frac{3696}{3} = \mathbf{1232 \text{ cm}^3}

Why This Works

The CSA formula $\pi r l$ comes from “unrolling” the curved surface of the cone. When you cut along the slant height and flatten it out, you get a sector of a circle with radius $l$ (the slant height) and arc length $2\pi r$ (the circumference of the base). The area of this sector works out to $\pi r l$ .

The volume formula $\frac{1}{3}\pi r^2 h$ tells us a cone is exactly one-third of a cylinder with the same base and height. This $\frac{1}{3}$ factor can be proven using calculus (integration of circular cross-sections), but for now, think of it as: three identical cones could fill one cylinder.

Alternative Method — Finding l when given r and h

If the problem gives $r$ and $h$ instead, compute the slant height first:

l = \sqrt{r^2 + h^2}

Then proceed with CSA and TSA as above. The volume doesn’t need $l$ at all — just $r$ and $h$ .

In CBSE, when the problem says “total surface area”, they want the base included. When it says “lateral surface area” or “curved surface area”, they don’t. Read the question carefully — 1 mark can be lost on this distinction alone.

Common Mistake

The biggest trap: using $h$ (height) in the CSA formula instead of $l$ (slant height). CSA $= \pi r l$ , NOT $\pi r h$ . Height and slant height are different — the slant height is always longer. If you confuse them, your CSA will be smaller than the actual value.

Question

Solution — Step by Step

Why This Works

Alternative Method — Finding l when given r and h

Common Mistake

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