Volume and Surface Area

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What: Volumes and surface areas of cube, cuboid, cylinder, cone, sphere, hemisphere, and frustum. Also melting and recasting problems where volume is conserved.
Why it matters: CDS averages 4–7 questions per paper on this single chapter — the largest mensuration head.
Key fact: When a solid is melted and recast, the volume is conserved. Set old volume = new volume; solve for the unknown dimension. This identity covers nearly every "sphere melted into cone" type CDS problem.

Volume and Surface Area is a formula-driven chapter where memorising the dozen standard formulas wins almost every mark. CDS reinforces the same archetypes: dimensions given, find volume; melt-and-recast for new dimension; ratio of two solids' volumes or surface areas.

This page is built from CDS Previous Year Questions across 2000–2023, plus NCERT Class 10 Surface Areas and Volumes. Pair with Area and Perimeter.

What This Topic Covers

CDS scope: (1) cube and cuboid; (2) cylinder (solid and hollow); (3) cone (volume and slant height); (4) sphere and hemisphere; (5) frustum of cone; (6) combinations — hemisphere on cylinder, cone on cylinder; (7) melt-and-recast; and (8) percentage change in volume / surface area when dimensions change.

Why This Topic Matters

4–7 CDS questions per paper.
Formula-driven — pure memorisation wins.
Melt-and-recast is a one-line equation (volume in = volume out).

Exam Pattern & Weightage

Year / Paper	No.	Subtopics Tested
2007-II	4	Cylinder, cube
2008-I/II	4	Sphere, cone
2010-I/II	5	Melt and recast, combination
2011-I/II	4	Cylinder hollow, frustum
2012-I/II	5	Sphere into cone, cube
2013-I/II	5	Hemisphere, cuboid
2014-I/II	5	Combination solids
2015-I/II	5	Mixed
2016-I/II	4	Cylinder, sphere ratio
2017-I/II	5	Frustum, melt-recast
2018-I/II	5	Mixed
2019-II	4	Cone, sphere
2020-I/II	5	Mixed
2021-I/II	5	Percentage change in volume
2022-I / 2023-I	6	Mixed

⚡ CDS Alert

If a sphere of radius $R$ is melted into $n$ smaller spheres each of radius $r$, volume conservation gives $\tfrac{4}{3}\pi R^3 = n \cdot \tfrac{4}{3}\pi r^3 \implies R^3 = n r^3 \implies n = (R/r)^3$. The 4/3 and $\pi$ cancel — a clean one-line answer.

Core Concepts

Cube and Cuboid

Cube (edge $a$) Volume $= a^3$; Total Surface Area (TSA) $= 6a^2$; Diagonal $= a\sqrt{3}$.

Cuboid ($\ell, b, h$) Volume $= \ell b h$; TSA $= 2(\ell b + b h + h \ell)$; Diagonal $= \sqrt{\ell^2 + b^2 + h^2}$.

Cylinder

Solid Cylinder (radius $r$, height $h$) Volume $= \pi r^2 h$; Curved Surface Area (CSA) $= 2\pi r h$; TSA $= 2\pi r(r + h)$.

Cone

Cone (radius $r$, height $h$, slant $\ell$) Slant height $\ell = \sqrt{r^2 + h^2}$; Volume $= \tfrac{1}{3}\pi r^2 h$; CSA $= \pi r \ell$; TSA $= \pi r(r + \ell)$.

Sphere and Hemisphere

Sphere (radius $r$) Volume $= \tfrac{4}{3}\pi r^3$; Surface Area $= 4\pi r^2$.

Hemisphere (radius $r$) Volume $= \tfrac{2}{3}\pi r^3$; CSA $= 2\pi r^2$; TSA $= 3\pi r^2$.

Frustum of Cone

Frustum (radii $R, r$ and height $h$) Volume $= \tfrac{1}{3}\pi h (R^2 + r^2 + Rr)$; Slant $= \sqrt{h^2 + (R - r)^2}$; CSA $= \pi (R + r) \ell$.

Volume Conservation (Melt and Recast)

When a solid is melted and recast into a different shape, the volume is conserved (assuming no waste).

Conservation $$V_{\text{old}} = V_{\text{new}} \implies n V_{\text{small}} = V_{\text{big}}$$

Percentage Change in Volume

If all dimensions of a solid scale by $k$, volume scales by $k^3$ and surface area by $k^2$. For percentage changes, use successive-percentage logic on each dimension.

Worked Examples

Example 1 — Cylinder Volume (2007-II)

Q: Find the volume of a cylinder with radius 7 cm and height 10 cm.

Volume = $\pi r^2 h = \tfrac{22}{7} \cdot 49 \cdot 10 = 22 \cdot 70 = 1540$ cm³.

Example 2 — Sphere Melted into Spheres (2010-II)

Q: A sphere of radius 6 cm is melted to form spheres of radius 2 cm. How many such small spheres?

$n = (R/r)^3 = (6/2)^3 = 27$.

Example 3 — Cone TSA (2008-I)

Q: Find the total surface area of a cone with radius 7 cm and slant 10 cm.

TSA = $\pi r(r + \ell) = \tfrac{22}{7} \cdot 7 \cdot (7 + 10) = 22 \cdot 17 = 374$ sq cm.

Example 4 — Cube to Sphere (2012-II)

Q: A cube of edge 12 cm is melted to form a sphere. Find the sphere's radius.

$12^3 = \tfrac{4}{3}\pi r^3 \implies 1728 = \tfrac{4}{3}\pi r^3 \implies r^3 = 1296/\pi \approx 412.5$.
$r \approx 7.44$ cm.

Example 5 — Hemisphere Combined (2014-II)

Q: A solid is in the shape of a cylinder with a hemisphere on top, both with radius 7 cm and cylinder height 10 cm. Find the total volume.

Cylinder vol = $\pi \cdot 49 \cdot 10 = 490\pi$. Hemisphere vol = $\tfrac{2}{3}\pi \cdot 343 = \tfrac{686\pi}{3}$.
Total = $490\pi + \tfrac{686\pi}{3} = \tfrac{1470\pi + 686\pi}{3} = \tfrac{2156\pi}{3}$ cm³.

Example 6 — Frustum (2017-I)

Q: Find the volume of a frustum with radii 6 cm and 4 cm, height 12 cm.

$V = \tfrac{1}{3}\pi h (R^2 + r^2 + R r) = \tfrac{1}{3}\pi \cdot 12 \cdot (36 + 16 + 24) = 4\pi \cdot 76 = 304\pi$ cm³.

Example 7 — Percentage Change in Volume (2021-I)

Q: If the radius of a sphere is increased by 10%, find the percentage increase in its volume.

Volume scales as $r^3$. New volume = (1.10)³ × old = 1.331 × old.
Percentage increase = 33.1%.

How CDS Tests This Topic

Six recurring archetypes: (1) basic volume or surface area given dimensions, (2) melt-and-recast — volume conservation, (3) ratio of volumes or surface areas of two solids, (4) combination solid (cone on cylinder, hemisphere on cylinder), (5) frustum problems, (6) percentage change in volume when a dimension changes.

Exam Shortcuts (Pro-Tips)

Shortcut 1 — Memorise the Dozen Formulas

Volume and surface area for cube, cuboid, cylinder, cone, sphere, hemisphere. Twelve formulas — six shapes × two each.

Shortcut 2 — Melt and Recast One-Liner

$V_1 = V_2$ ⇒ cancel common factors of $\pi$ and constants → linear or cubic equation in unknown dimension.

Shortcut 3 — Cube of Linear Ratio

If two similar solids have dimensions in ratio $a:b$, their volumes are in ratio $a^3 : b^3$ and surface areas $a^2 : b^2$. Memorise.

Shortcut 4 — $\pi = 22/7$

CDS questions are usually designed so the $22/7$ approximation cancels cleanly with dimensions involving 7, 14, 21, 28, etc. Look for these multipliers and use $\pi = 22/7$.

Shortcut 5 — Diagonal of Cube

Cube edge $a$: face diagonal $a\sqrt{2}$, space diagonal $a\sqrt{3}$. Cuboid space diagonal $\sqrt{\ell^2 + b^2 + h^2}$.

Common Question Patterns

Pattern 1 — Find Volume / SA Given Dimensions

Direct formula application.

Pattern 2 — Melt and Recast

Set old volume = new volume. Solve.

Pattern 3 — Ratio of Two Solids

Cube the linear ratio for volume; square for surface area.

Pattern 4 — Combination Solid

Sum volumes (or surface areas, accounting for hidden bases).

Pattern 5 — Percentage Change

Identify scaling factor; apply $k^3$ for volume, $k^2$ for surface area.

Preparation Strategy

Week 1. Memorise all six standard solids' formulas. Drill 20 basic problems. Practice $\pi = 22/7$ recognition.

Week 2. Melt-and-recast, combination solids, frustum, percentage change. Cross-train with Area and Perimeter.

Mock testing. Use CDS mock tests. Most slip-ups: forgetting the $1/3$ for cones and pyramids, or confusing CSA and TSA in cylinders. Drill both.

Drill Volume and Surface Area

CDS mocks with melt-and-recast, combinations, frustums, and percentage change. Six archetypes — twelve formulas — reflex.

Start Free Mock Test

Frequently Asked Questions

What's the volume formula for a sphere?

$V = \tfrac{4}{3}\pi r^3$. Surface area = $4\pi r^2$. For a hemisphere, halve the volume and adjust the surface area to $3\pi r^2$ (total includes the flat circular base).

How does melt-and-recast work?

Volume is conserved. Set old solid's volume = new solid's volume (or $n \times$ volume of $n$ new solids). Solve for the unknown dimension.

What's the slant height of a cone?

For a cone with radius $r$ and height $h$: slant $\ell = \sqrt{r^2 + h^2}$. The CSA of the cone is $\pi r \ell$; TSA = $\pi r(r + \ell)$.

How do I find the volume of a frustum?

For a frustum with bottom radius $R$, top radius $r$, and height $h$: $V = \tfrac{1}{3}\pi h (R^2 + r^2 + R r)$. The slant height is $\sqrt{h^2 + (R - r)^2}$ and CSA = $\pi(R + r)\ell$.

If I scale all dimensions by $k$, how do volume and surface area change?

Volume scales by $k^3$; surface area by $k^2$. Example: doubling all dimensions multiplies volume by 8 and surface area by 4.

How do I handle combination solids?

Decompose into known solids. Volume is just the sum. Surface area requires care — when two solids share a face, that face is "inside" and not counted in the total surface area.

Which CDS Maths topics connect to Volume and Surface Area?

Area and Perimeter — the 2D parent. Percentage — for dimension-change problems. Triangles — for cones and frustums.