Surface Areas and Volumes — Detailed Notes

Chapter 12 | CBSE Class 10 Mathematics (NCERT)

Study Diagrams (Visual Revision)

Use these board-style figures while reading the chapter. Open the Diagrams tab for a larger gallery.

Surface Areas & Volumes — Decision Map

Core Solids — Visual Summary

Surface Areas & Volumes — Formula Strip

12.1 Introduction

From Class IX: cuboid, cone, cylinder, sphere. Many real objects are combinations of these (oil tanker = cylinder + two hemispheres; test tube = cylinder + hemisphere). This chapter finds surface areas and volumes of such combined solids.

Recall (Class 9 toolkit)

Solid	CSA / curved	TSA	Volume
Cuboid	—	$2 (l b + bh + h l)$	$l bh$
Cylinder	$2 π r h$	$2 π r (h + r)$	$π r^{2} h$
Cone	$π r l$	$π r (l + r)$	$\frac{1}{3} π r^{2} h$
Sphere	$4 π r^{2}$	$4 π r^{2}$	$\frac{4}{3} π r^{3}$
Hemisphere	$2 π r^{2}$	$3 π r^{2}$	$\frac{2}{3} π r^{3}$

12.2 Surface Area of a Combination of Solids

Guiding principle

When solids are joined, common/joining faces are no longer part of the outer surface. So:

$TSA(new solid) \neq = TSA(part_{1}) + TSA(part_{2})$

Instead, add only the exposed curved/flat surfaces (and subtract covered areas when a solid sits on another).

Pattern A — Cylinder with hemispherical ends (tanker / capsule)

Visible: CSA of cylinder + CSA of two hemispheres ( $=$ CSA of one full sphere $4 π r^{2}$ if radii match).

$TSA = 2 π r h + 4 π r^{2} = 2 π r (h + 2 r)$ (Here $h$ = cylindrical length only.)

Pattern B — Cone surmounted by hemisphere (toy / lattu)

$TSA = CSA(hemisphere) + CSA(cone) = 2 π r^{2} + π r l$ Flat bases cancel (joined and hidden).

Pattern C — Cube with hemisphere on top

$Surface = TSA(cube) - π r^{2} + 2 π r^{2} = TSA(cube) + π r^{2}$ (Base of hemisphere covers $π r^{2}$ of the cube but adds curved $2 π r^{2}$ .)

Pattern D — Different radii (cone on thinner cylinder / rocket)

Include the exposed ring on the cone’s base: $π R^{2} - π r^{2}$ .

Better Method / Strategy — Surface area

Sketch and shade only outer faces.

Write CSA/TSA of each part.

Subtract every joined circle/square.

If radii differ, add the annular ring.
Never blindly sum Class-9 TSAs.

Worked ideas (NCERT-aligned)

Ex. 1 style — Playing top: cone + hemisphere, total height $5$ cm, diameter $3.5$ cm.
$r = \frac{3.5}{2}$ , cone height $= 5 - r = 3.25$ , $l = r^{2} + h^{2} \approx 3.7$ .
TSA $= 2 π r^{2} + π r l \approx 39.6$ cm².

Ex. 2 style — Cube + hemisphere: edge $5$ cm, hemisphere diameter $4.2$ cm.
Surface $= 150 + π r^{2} = 163.86$ cm².

Ex. 3 style — Rocket: cone ( $r = 2.5$ , $h = 6$ ) on cylinder ( $r^{'} = 1.5$ , $h^{'} = 20$ ).
Orange (cone side): $π r l + π r^{2} - π (r^{'})^{2}$ .
Yellow (cylinder): $2 π r^{'} h^{'} + π (r^{'})^{2}$ .

Ex. 4 style — Bird-bath: cylinder with hemispherical depression.
TSA $= 2 π r h + 2 π r^{2} = 2 π r (h + r)$ .

12.3 Volume of a Combination of Solids

Guiding principle

Joining does not remove volume. Volumes add. Cavities/depressions subtract.

$V_{combined} = V_{1} + V_{2} + \dots or V_{remaining} = V_{outer} - V_{cavity}$

Better Method / Strategy — Volume
Surface area cares about skin; volume cares about capacity.
Apparent capacity of a glass $=$ cylinder volume; actual $=$ cylinder $-$ raised hemisphere.

Worked ideas

Ex. 5 style — Shed: cuboid + half-cylinder on top.
$V = l bh + \frac{1}{2} π r^{2} L$ , then subtract space of machinery/workers if asked.

Ex. 6 style — Juice glass: apparent $= π r^{2} h$ ; actual $= π r^{2} h - \frac{2}{3} π r^{3}$ .

Ex. 7 style — Hemisphere + cone toy:
$V = \frac{2}{3} π r^{3} + \frac{1}{3} π r^{2} h$ .
If a cylinder circumscribes it, difference $= π r^{2} H - V_{toy}$ .

Common combination volumes

Object	Volume idea
Capsule	$π r^{2} h + \frac{4}{3} π r^{3}$ (cylinder + sphere)
Cone on hemisphere	$\frac{1}{3} π r^{2} h + \frac{2}{3} π r^{3}$
Gulab jamun shape	cylinder + 2 hemispheres
Pen stand	cuboid $-$ conical holes
Water displaced	volume of immersed solid

More surface-area patterns (board frequent)

Tent (cylinder + conical top, no floor): canvas $= 2 π r h + π r ℓ = π r (2 h + ℓ)$ .

Hemisphere scooped from each end of a cylinder:
outer CSA of cylinder remains, plus two inner hemispherical CSAs (and check ends): typically
$2 π r h + 2 \cdot 2 π r^{2} = 2 π r (h + 2 r)$ when flat ends are replaced by open hemi cavities — follow the figure carefully.

Conical cavity in a cylinder (same $r, h$ ):
TSA $=$ CSA(cylinder) $+$ base of cylinder $+$ CSA(cone) $= π r (2 h + r + ℓ)$ .

Medicine capsule: treat as cylinder of length $L - 2 r$ plus full sphere surface $4 π r^{2}$ .

More volume patterns

Gulab jamun (cylinder + 2 hemispheres): like capsule volume; syrup fraction of volume as given.

Lead shots in a conical vessel: volume of water overflow $= \frac{1}{4}$ of cone $=$ $n \times \frac{4}{3} π R^{3}$ → solve for $n$ .

Solid in a water-filled cylinder: water left $=$ cylinder volume $-$ immersed solid volume.

Pen stand: $V_{wood} = l bh - 4 \cdot \frac{1}{3} π r^{2} h_{cone}$ .

Better Method / Strategy — Combination solids
Ask two questions every time:
(1) What is visible? → surface
(2) What space is occupied/empty? → volume
Sketch a “before joining / after joining” diagram for 10 seconds — prevents most TSA errors.

Unit and $π$ hygiene

Convert all dimensions to the same unit before substituting.
$1$ m $= 100$ cm $\Rightarrow$ $1$ m² $= 1 0^{4}$ cm², $1$ m³ $= 1 0^{6}$ cm³.
If mass density is per cm³, keep volume in cm³.
Do not mix $π = \frac{22}{7}$ mid-solution with $3.14$ unless the question switches.

NCERT summary goals

Determine surface area of an object formed by combining any two of: cuboid, cone, cylinder, sphere, hemisphere.
Find volume of objects formed by combining any two of those solids.

Exam-Oriented Analysis

Question type	Typical marks	Key skill
Toy / lattu / capsule TSA	3–4	Exposed CSA only
Tent canvas + cost	3–4	Exclude floor; multiply rate
Cube + hemisphere / depression	3	$\pm π r^{2}$ correctly
Vessel / glass capacity	3–4	Add/subtract volumes
Pole mass / water left	4–6	Multi-step volume

Common mistakes

Adding TSAs of parts without removing joined faces
Using hemisphere TSA $3 π r^{2}$ when only CSA $2 π r^{2}$ is exposed
Forgetting slant height for cone CSA
Mixing diameter/radius; cm/m conversion errors
Using volume formula when “paint/canvas” asks for surface area
Capsule: taking cylinder height as full length including hemispheres
Ignoring the annular ring when cone base radius $>$ cylinder radius

Scoring tips

Start with a one-line decomposition: “solid = A + B − cavity C”
Compute $l = r^{2} + h^{2}$ before cone CSA
Keep $π = \frac{22}{7}$ or $3.14$ as specified; show substitution
Box answer with correct unit (cm² vs cm³)
For two-colour painting (rocket style), split the surface into named parts and total each colour separately