Surface Areas & Volumes | Maths Grade 10 CBSE

Play with it

Resize a solid

Pick a shape, then drag the radius and height. The volume and surface areas recompute from their formulas — see how a cone is exactly a third of its cylinder.

radius r 5

height h 9

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The three ideas in this chapter

Keep this table handy — every problem starts here:

Solid	Volume	Curved SA	Total SA
Cylinder	πr²h	2πrh	2πr(r+h)
Cone	⅓πr²h	πrl	πr(l+r)
Sphere	(4/3)πr³	4πr²	4πr²
Hemisphere	(2/3)πr³	2πr²	3πr²

For a cone, the slant height l = √(r² + h²) (Pythagoras). Volume is in cubic units; surface area in square units.

Common mistake: using h instead of the slant l in a cone's curved surface area. CSA = πrl, not πrh.

A cone and a cylinder with the same base and height are linked: it takes exactly three cone-fulls to fill the cylinder. That's the (1/3) in the cone's volume.

Worked example · cylinder volume

Find the volume of a cylinder with r = 7 cm, h = 10 cm (take π = 22/7).

V = πr²h = (22/7)·7²·10.
= (22/7)·49·10 = 22·70.
= 1540 cm³.

Quick check

A cone with the same base and height would hold 1540 ÷ 3 ≈ 513.3 cm³ — a third, exactly as the formula says.

Real objects are solids stuck together — an ice-cream (cone + hemisphere), a capsule (cylinder + 2 hemispheres), a tent (cylinder + cone). To measure them:

Volume: just add the parts' volumes.
Surface area: add only the outer faces — the joining circle is inside, so don't count it.

Worked example · ice-cream volume

An ice-cream is a cone (r = 3.5 cm, h = 12 cm) topped by a hemisphere of the same radius (π = 22/7).

Cone = (1/3)·(22/7)·3.5²·12 = (1/3)·(22/7)·12.25·12 = 154 cm³.
Hemisphere = (2/3)·(22/7)·3.5³ = (2/3)·(22/7)·42.875 ≈ 89.83 cm³.
Total ≈ 243.83 cm³.

Common mistake: double-counting the joined face in surface area. The circle where two solids meet is hidden inside — leave it out.

Why this matters

Where you'll actually use this

Anyone who fills, coats, ships or builds a 3-D thing needs this: how much fits inside (volume) and how much surface to cover (area). Tanks, packaging, fuel, paint — all of it.

Tanks, bottles & storage

How many litres does a cylindrical water tank hold? That's its volume, πr²h. Bottlers, fuel depots and overhead tanks are all sized this way — and the surface area tells the maker how much steel or plastic to use.

Volume = capacity

Packaging & design

An ice-cream, a medicine capsule, a circus tent, a grain silo — designers model each as solids joined together, then add volumes for capacity and add outer areas for material. Getting it right saves real money on every unit made.

Combinations of solids

🎨 Paint & coating

Surface area decides how much paint, plating or wrap a 3-D object needs.

⛽ Fuel & tankers

Cylindrical and capsule tankers are sized by volume to carry an exact load.

🏗️ Domes & silos

Architects compute hemispherical domes and conical roofs for material and capacity.

🔒 More real-world applications

Paint estimates, tankers, domes and more — each explained with a diagram. Free to unlock.

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Competency quiz

Modelled on CBSE's competency pattern — MCQ, assertion–reason and case-study items.

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