Given,

Radius of the conical vessel (r) = 7 cm

Slant height of the conical vessel (l) = 25 cm

Let height of cone = h cm

We know that,

⇒ l² = r² + h²

⇒ h² = l² - r²

⇒ h = $\sqrt{l^2 - r^2}$

Substituting values we get :

$\Rightarrow h = \sqrt{(25)^2 - (7)^2} \\[1em] \Rightarrow h = \sqrt{625 - 49} \\[1em] \Rightarrow h = \sqrt{576} = 24 \text{ cm}.$

By formula,

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

Substituting values we get :

$V = \dfrac{1}{3} \times \dfrac{22}{7} \times 7^2 \times 24 \\[1em] = \dfrac{1}{3} \times \dfrac{22}{7} \times 49 \times 24 \\[1em] = 22 \times 7 \times 8 \\[1em] = 1232 \text{ cm}^3.\\[1em] = 1232 \times \Big(\dfrac{1}{1000}\Big) \text{ l} \quad [∵ 1000 \text{ cm}^3 = 1 \text{ litre] }\\[1em] = \text{1.232} \text{ l}.$

Hence, capacity of the conical vessel = 1.232 l.

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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)