Given,

Height of the conical vessel (h) = 12 cm

Slant height of the conical vessel (l) = 13 cm

Let radius of the vessel = r cm

We know that,

⇒ l² = r² + h²

⇒ r² = l² - h²

⇒ r = $\sqrt{l^2 - h^2}$

Substituting values we get :

$r = \sqrt{(13)^2 - (12)^2} \\[1em] = \sqrt{169 - 144} \\[1em] = \sqrt{25} = 5 \text{ cm}$

By formula,

Capacity of the conical vessel (V) = $\dfrac{1}{3}πr^2h$

Substituting values we get :

$V = \dfrac{1}{3} \times \dfrac{22}{7} \times 5^2 \times 12 \\[1em] = \dfrac{1}{3} \times \dfrac{22}{7} \times 300 \\[1em] = \dfrac{6600}{21} \\[1em] = \dfrac{2200}{7} \\[1em] = \dfrac{2200}{7} \times \dfrac{1}{1000} \text{ l} \quad [∵ 1000 \text{ cm}^3 = 1 l] \\[1em] = \dfrac{22}{70} \text{ l} \\[1em] = \dfrac{11}{35} \text{ l}.$

Hence, capacity of the conical vessel = $\dfrac{11}{35}$ l.

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CBSE Class IX | Academic Year 2026-2027

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Surface Areas and Volumes — Interactive Study Guide

Formula Master Table

Solid	CSA	TSA	Volume
Cube (a)	4a²	6a²	a³
Cuboid (l,b,h)	2h(l+b)	2(lb+bh+hl)	lbh
Cylinder (r,h)	2πrh	2πr(r+h)	πr²h
Cone (r,h,l)	πrl	πr(l+r)	⅓πr²h
Sphere (r)	4πr²		&frac43;πr³
Hemisphere (r)	2πr²	3πr²	⅔πr³

Common Traps

Cone CSA uses slant height l, NOT height h! l = √(r²+h²)

Hemisphere TSA = CSA + base circle = 2πr² + πr² = 3πr²

Use radius, not diameter! If diameter is given, divide by 2 first.

Quick Self-Check

1. Volume of cube with side 5 cm? (125 cm³)
2. CSA of cylinder with r=7, h=10? (2×22/7×7×10 = 440 cm²)
3. Volume of cone with r=3, h=4? (⅓×π×9×4 = 12π cm³)