Surface Areas and Volumes notes: CBSE Class 10 Mathematics (Standard 041 / Basic 241)

1. The six basic solids

Solid	Curved / lateral surface	Total surface	Volume
Cuboid	$2h(l + b)$	$2(lb + bh + hl)$	$lbh$
Cube	$4a^2$	$6a^2$	$a^3$
Cylinder	$2\pi rh$	$2\pi r(r + h)$	$\pi r^2h$
Cone	$\pi rl$	$\pi r(l + r)$	$\frac13\pi r^2h$
Sphere	—	$4\pi r^2$	$\frac43\pi r^3$
Hemisphere	$2\pi r^2$	$3\pi r^2$	$\frac23\pi r^3$

Use the value of $\pi$ the question gives. Not in the 2026–27 syllabus: conversion of one solid into another (melting, recasting, digging) and the frustum of a cone.

Learn this table: every combined solid is built from it.
Solid	Curved / lateral surface	Total surface	Volume
Cuboid $l \times b \times h$	$2h(l + b)$	$2(lb + bh + hl)$	$lbh$
Cube, edge $a$	$4a^2$	$6a^2$	$a^3$
Cylinder	$2\pi rh$	$2\pi r(r + h)$	$\pi r^2h$
Cone	$\pi rl$	$\pi r(l + r)$	$\dfrac13\pi r^2h$
Sphere	—	$4\pi r^2$	$\dfrac43\pi r^3$
Hemisphere	$2\pi r^2$	$3\pi r^2$	$\dfrac23\pi r^3$

The building blocks of every combined solid in this chapter.

Key terms

Curved surface area (CSA): The area of the curved part only, without flat faces.
Total surface area (TSA): Curved surface plus all flat faces of a single solid.

2. The cone and its slant height

For a cone of radius $r$ and vertical height $h$ , the slant height is

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

Use $l$ in the curved surface $\pi rl$ and $h$ in the volume $\dfrac13\pi r^2h$ . Useful triples: $(5, 12, 13)$ , $(7, 24, 25)$ , $(21, 20, 29)$ , $(3.5, 12, 12.5)$ , $(2.8, 2.1, 3.5)$ .

A cone and a cylinder with the same base and height: the cone has one-third of the volume.

Slant height: $l = \sqrt{r^2 + h^2}$.

Formulas

Slant height
$l = \sqrt{r^2 + h^2}$

Key terms

Slant height: The distance from the apex of a cone to any point on the edge of its base.

3. Surface area of a combination

Rule: add the visible surfaces only. Faces in contact are inside the solid; open faces (a tent's floor, a cup's top) are not there.

Tent (cylinder $+$ cone, no floor): $2\pi rh + \pi rl$ .
Kulfi or toy (cone $+$ hemisphere): $\pi rl + 2\pi r^2$ .
Capsule or tanker (cylinder $+$ two hemispheres): $2\pi rh + 4\pi r^2$ .
Rocket (cylinder $+$ cone, with bottom): $2\pi rh + \pi r^2 + \pi rl$ .
Cube with a cube or cuboid on top: the covered square reappears as the top of the upper block, so add only its side faces.

Joining two solids with a common circle of radius $r$ removes $2\pi r^2$ from the sum of their total surfaces.

Add the surfaces you can touch from outside; drop the joined faces.
Combined solid	Hidden faces	Surface area
Cone on cylinder (tent, no floor)	joining circle, floor	$2\pi rh + \pi rl$
Hemisphere on cone (kulfi)	joining circle	$\pi rl + 2\pi r^2$
Cylinder with two hemispherical ends	both end circles	$2\pi rh + 4\pi r^2$
Hemisphere scooped from a cube	circle cut from the face	$6a^2 - \pi r^2 + 2\pi r^2$

Tent: cylinder $+$ cone; the canvas has no floor.

Cylinder $+$ two hemispheres: surface $= 2\pi rh + 4\pi r^2$.

Formulas

Cone on a hemisphere
$S = \pi r l + 2\pi r^2$
Cylinder with two hemispherical ends
$S = 2\pi r h + 4\pi r^2$

4. Scooping out and sticking on

Scooping a hemisphere of radius $r$ from a face: the face loses $\pi r^2$ , the bowl adds $2\pi r^2$ — surface increases by $\pi r^2$ ; volume decreases by $\dfrac23\pi r^3$ .
Sticking a hemisphere on a face: the face loses $\pi r^2$ (covered), the dome adds $2\pi r^2$ — surface increases by $\pi r^2$ ; volume increases by $\dfrac23\pi r^3$ .
Hollowing a cone from a cylinder (same $r$ , $h$ ): the top circle goes, the inside of the cone $\pi rl$ appears; volume left $= \dfrac23\pi r^2h$ .

Example: cube of edge $10$ cm with a hemisphere of radius $3$ cm scooped out: surface $600 + 28.26 = 628.26$ cm².

Scooping removes a circle $\pi r^2$ from the top but adds the bowl $2\pi r^2$.

Cylinder minus a cone with the same base and height.

Formulas

Scooping a hemisphere
$\Delta S = -\pi r^2 + 2\pi r^2 = +\pi r^2$

5. Volume of a combination

Rule: joined parts add; removed parts subtract.

Cylinder $+$ cone on the same base: $\pi r^2\left(h_1 + \dfrac{h_2}{3}\right)$ (silo $r = 3.5$ , $h_1 = 10$ , $h_2 = 3$ : $423.5$ m³).
Cone $+$ hemisphere: $\dfrac13\pi r^2(h + 2r)$ (kulfi $r = 3.5$ , $h = 12$ : $\approx 243.83$ cm³).
Cylinder $+$ hemisphere: $\pi r^2\left(h + \dfrac{2r}{3}\right)$ .
Block with a hollow: block $-$ hollow.

When the total height of a vessel with a hemispherical bottom is given, the cylinder's height is total $-$ $r$ . The smallest box around a solid has the same radius and the full height.

Volume: add what is joined, subtract what is removed.

Silo: volume $=$ cylinder $+$ cone.

Tank: cylinder $+$ hemispherical bottom.

Formulas

Cone on a hemisphere
$V = \frac13\pi r^2(h + 2r)$
Cylinder with a cone on top
$V = \pi r^2\left(h_1 + \frac{h_2}{3}\right)$

6. Units and capacity

$1$ m³ $= 1000$ litres; $1$ litre $= 1000$ cm³ $= 1000$ mL; $1$ m³ $= 10^6$ cm³.
Capacity is the volume of the inside of a container.
Cost $=$ area $\times$ rate (per m² or per cm²) or volume $\times$ rate; keep units consistent.

Example: tank $74.844$ m³ $= 74844$ litres.

$1$ m³ $= 1000$ L and $1$ L $= 1000$ cm³.
From	To	Multiply by
m³	litres	$1000$
litres	cm³	$1000$
m³	cm³	$1000000$
cm³	litres	$\dfrac{1}{1000}$

Formulas

Litres
$1\ \text{m}^3 = 1000\ \text{L}$

Key terms

Capacity: The volume a container can hold, often in litres.

Exam tips

How marks are awarded

Surface area (2–3 marks): slant height 1; visible faces named 1; answer with unit 1.
Volume (2–3 marks): each part 1; sum or difference with unit 1.
Long questions (5 marks): typically slant height, surface, cost, volume and a conversion, one mark each.

Common mistakes

Slant height in a volume, or vertical height in $\pi rl$ .
$3\pi r^2$ for a hemisphere that is glued to another solid.
Adding total surface areas of the parts.
Forgetting $\dfrac13$ (cone) or $\dfrac23$ (hemisphere).
Using the total height as the cylinder's height.
Litres: multiply m³ by $1000$ , divide cm³ by $1000$ .

1. The six basic solids

Key terms

2. The cone and its slant height

Formulas

Key terms

3. Surface area of a combination

Formulas

4. Scooping out and sticking on

Formulas

5. Volume of a combination

Formulas

6. Units and capacity

Formulas

Key terms

Exam tips

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