Volume of cone = (1/3)πr²h. Substitute r=7, h=24 to get 1232 cm³.

Find Volume of a Cone with Radius 7 cm, Height 24 cm

Question

Find the volume of a cone with radius 7 cm and height 24 cm. (NCERT Class 9, Chapter 13)

Solution — Step by Step

The volume of a cone is:

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

The $\frac{1}{3}$ factor is what separates cones from cylinders — a cone holds exactly one-third the volume of a cylinder with the same base and height.

From the problem:

Radius $r = 7$ cm
Height $h = 24$ cm

We use $\pi = \frac{22}{7}$ here since $r = 7$ — this cancels the 7 cleanly and gives us a whole-number answer.

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

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

Cancel $7$ from $\frac{22}{7}$ and $49$ :

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

V = \frac{1}{3} \times 3696 = 1232 \text{ cm}^3

V = 1232 \text{ cm}^3

Why This Works

The formula $V = \frac{1}{3}\pi r^2 h$ can be understood by comparison. A cylinder with the same base (radius $r$ ) and same height $h$ has volume $\pi r^2 h$ . A cone with the same dimensions holds exactly one-third of that — this is a provable result from integral calculus, but for now, trust the geometry.

The choice of $\pi = \frac{22}{7}$ is deliberate here, not arbitrary. When the radius is a multiple of 7, the $\frac{22}{7}$ cancels beautifully. NCERT uses $r = 7$ in this problem precisely so the arithmetic stays clean.

Alternative Method — Decimal Approximation

If the exam allows $\pi \approx 3.14$ (some schools specify this):

V = \frac{1}{3} \times 3.14 \times 49 \times 24

V = \frac{1}{3} \times 3693.12 = 1231.04 \text{ cm}^3

NCERT and CBSE board exams almost always say “use $\pi = \frac{22}{7}$ ” unless stated otherwise. When $r$ is a multiple of 7, this gives exact whole-number answers — a strong signal you’re on the right track.

Common Mistake

The most common error: forgetting the $\frac{1}{3}$ . Students write $V = \pi r^2 h$ directly — which is the cylinder formula. This gives $3696$ cm³ instead of $1232$ cm³, exactly three times too large. If your answer is a multiple of 3 compared to the expected answer, this is why.

A second slip: squaring the height instead of the radius. The formula is $r^2 \times h$ , not $r \times h^2$ . Write $r$ and $h$ clearly before substituting so you don’t mix them up mid-calculation.

Question

Solution — Step by Step

Why This Works

Alternative Method — Decimal Approximation

Common Mistake

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