SSC CGL Maths · Mensuration 3D

Scaling and Comparing Solids

When every length of a solid is multiplied by k, its areas are multiplied by k² and its volume by k³; two different solids are compared by writing both formulas and cancelling what they share.

All Mensuration 3D notes

Why this matters

Twenty-three questions in the bank, fourteen EASY and nine MODERATE, nineteen of them from the 2025 shifts. None of them needs a full volume: each asks how many times, what ratio or what percentage, so you work with factors and π cancels. This is where a strong student saves the most time.

Concept 1 of 4: How a change in length changes area and volume

Double every length of a solid and each face becomes 2 × 2 = 4 times as big, while the solid holds 2 × 2 × 2 = 8 times as much. Areas follow the square of the scale and volumes follow the cube.

Definition

If every length is multiplied by $k$ , every area is multiplied by $k^2$ and every volume by $k^3$ .
If the lengths change by different factors, multiply the factors the way the formula does. A cone or a cylinder holds an amount that goes with $r^2 h$ , so radius $\times p$ and height $\times q$ give volume $\times p^2 q$ .
A percentage change is a factor: 15% more is $\times 1.15$ , 25% less is $\times 0.75$ . Turn the final factor back at the end: $\times 1.089$ is a rise of 8.9%.
"Increased by 200%" means $\times 3$ , not $\times 2$ . "The volume becomes 8 times" is an increase of 7 times, or 700%.
Fast route: the same $x\%$ rise on all three edges multiplies the volume by $(1 + \frac{x}{100})^3$ , so 10% on each edge gives $1.331$ , a 33.1% rise. Safe only when every edge gets the same rise.

Scale factors

\text{area} \times k^2 \qquad \text{volume} \times k^3 \qquad V_{\text{cone, cylinder}} \propto r^2 h

Worked example

The radius of a cylinder is increased by 30% and its height is decreased by 50%. Find the percentage change in its volume.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 3 · Q63Easy

Example 1 · Mensuration 3D · Scaling and Comparing Solids

The radius of a cone is raised by

20\%

, while the height is reduced by

10\%

. What is the percentage change in the volume of the cone?

Adding the percentages

Radius up 10% and height down 10% does not leave a cone's volume unchanged. The radius counts twice, as

r^2

1.1 \times 1.1 \times 0.9 = 1.089

, a rise of 8.9%.

Increased by 200% means three times

An increase of 200% adds twice the length to itself, so the new length is 3 times the old one, not 2 times.

New size or increase

If a volume becomes 8 times as large, the increase is 7 times the original, that is 700%. Options often print both 8 and 7, or 800% and 700%.

Drill 7 more on how a change in length changes area and volume

Concept 2 of 4: Ratios of two cylinders or two cones

Two cylinders differ only in their radius and their height, and so do two cones. Every ratio of their volumes or surfaces is built from the ratios of those two lengths.

Definition

Volumes: $\frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2}$ . The same holds for two cones, because the $\frac{1}{3}$ cancels.
Curved surfaces of two cylinders: $\frac{r_1 h_1}{r_2 h_2}$ .
For a missing ratio, solve the same equation: $\frac{h_1}{h_2} = \frac{V_1}{V_2} \times \frac{r_2^2}{r_1^2}$ .
This rule is for two solids of the same kind. A cone against a cylinder brings the $\frac{1}{3}$ back.

Two cylinders or two cones

\frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2} \qquad \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{r_1 h_1}{r_2 h_2}

Worked example

The radii of two cylinders are in the ratio

5 : 3

and their heights in the ratio

2 : 5

. Find the ratio of their volumes and the ratio of their curved surface areas.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 26 Sep 2025, Shift 3 · Q62Easy

Example 2 · Mensuration 3D · Scaling and Comparing Solids

The volumes of two right circular cylinders have a ratio of 2:3. If the radii of these cylinders are in the ratio of 1:2, what is the ratio of their heights?

Squaring the height too

Only the radius is squared in a volume ratio. With radii

1 : 2

and heights

1 : 2

the volumes are

1 \times 1 : 4 \times 2 = 1 : 8

, not

1 : 16

Drill 2 more on ratios of two cylinders or two cones

Concept 3 of 4: Slicing a cone by a plane parallel to the base

Cut a cone with a plane parallel to its base and the top piece is a small copy of the whole cone. Every length shrinks by the same factor, so the volumes go as the cube of that factor. The bottom piece, the frustum, is whatever is left.

Definition

The top cone is similar to the whole: if its height is $\frac{1}{k}$ of the whole height, its radius is $\frac{1}{k}$ of the base radius and its volume is $\frac{1}{k^3}$ of the whole.
Frustum $=$ whole cone $-$ top cone, that is $1 - \frac{1}{k^3}$ of the whole.
Measure heights from the tip. A cut 4 cm above the base of a 12 cm cone is 8 cm below the tip.
With several cuts, find each cone from the tip, then subtract neighbouring cones to get each piece.
Going back: a top cone holding $\frac{1}{8}$ of the volume has $\frac{1}{2}$ of the height (take the cube root).
The frustum also has its own formula, $\frac{1}{3}\pi h(R^2 + Rr + r^2)$ , but subtracting two cones is usually faster.

Cut cone and frustum

\frac{V_{\text{top}}}{V_{\text{whole}}} = \left(\frac{h}{H}\right)^3 \qquad V_{\text{frustum}} = \frac{1}{3}\pi h(R^2 + Rr + r^2)

$h, H$ height of the top cone and of the whole cone, both from the tip
$R, r$ radii of the two circular faces of the frustum

Worked example

A cone 20 cm tall is cut parallel to its base at 15 cm above the base. Find the ratio of the volume of the small top cone to the volume of the frustum.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 19 Sep 2025, Shift 3 · Q62Moderate

Example 3 · Mensuration 3D · Scaling and Comparing Solids

A right circular cone having height of 30 cm is cut by two parallel planes at heights 10 cm and 20 cm from the base. What is the ratio of the volumes of the three parts (from top to bottom)?

Measuring from the base

Similar-cone ratios use heights from the tip. A cut 6 cm above the base of a 24 cm cone leaves a top cone 18 cm tall, three-quarters of the height.

Height ratio used as the volume ratio

A top cone with half the height holds one-eighth of the volume, not one-half. Volumes go as the cube of the height ratio.

Drill 5 more on slicing a cone by a plane parallel to the base

Concept 4 of 4: Volume and surface formulas of the standard solids

To compare two different solids, write both formulas side by side, set equal whatever the question says is equal, and cancel π and the shared radius.

Definition

Two solids with the same radius and equal volumes: cancel $\pi r^2$ and compare what is left. A cone leaves $\frac{1}{3}h$ , a cylinder $h$ , a hemisphere $\frac{2}{3}r$ and a sphere $\frac{4}{3}r$ .
So a cone equals a cylinder on the same base when it is 3 times as tall, and equals a hemisphere on the same base when $h = 2r$ .
Two solids with equal surfaces: write both total surfaces from the table and cancel $\pi r$ .

Solid	Volume	Curved or side surface	Total surface
Cube, edge $a$	$a^3$	$4a^2$	$6a^2$
Cuboid	$lbh$	$2(l + b)h$	$2(lb + bh + hl)$
Prism	base area $\times h$	base perimeter $\times h$	side surface $+ 2 \times$ base area
Pyramid	$\frac{1}{3} \times$ base area $\times h$	$\frac{1}{2} \times$ base perimeter $\times$ slant height	side surface $+$ base area
Cylinder	$\pi r^2 h$	$2\pi rh$	$2\pi r(r + h)$
Cone	$\frac{1}{3}\pi r^2 h$	$\pi rl$	$\pi r(l + r)$
Sphere	$\frac{4}{3}\pi r^3$	$4\pi r^2$	$4\pi r^2$
Solid hemisphere	$\frac{2}{3}\pi r^3$	$2\pi r^2$	$3\pi r^2$ The total surface includes the flat circle; a bowl has no lid, so it has only the curved part.

In the cone row

l

is the slant height; in the cuboid row

l

is the length.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 3 · Q60Moderate

Example 4 · Mensuration 3D · Scaling and Comparing Solids

A cone and a sphere have equal volumes. If the cone's height is twice its radius, what is the ratio of the cone's radius to the sphere's radius?

Cone against cylinder the wrong way round

A cone holds one third of the cylinder on the same base and height. For equal volumes the cone must be 3 times as tall as the cylinder, not one third as tall.

Drill 5 more on volume and surface formulas of the standard solids

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

How a change in length changes area and volume
Scale factors
$\text{area} \times k^2 \qquad \text{volume} \times k^3 \qquad V_{\text{cone, cylinder}} \propto r^2 h$
Ratios of two cylinders or two cones
Two cylinders or two cones
$\frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2} \qquad \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{r_1 h_1}{r_2 h_2}$
Slicing a cone by a plane parallel to the base
Cut cone and frustum
$\frac{V_{\text{top}}}{V_{\text{whole}}} = \left(\frac{h}{H}\right)^3 \qquad V_{\text{frustum}} = \frac{1}{3}\pi h(R^2 + Rr + r^2)$

Reference tables (1)

Volume and surface formulas of the standard solids8 rows

Solid	Volume	Curved or side surface	Total surface
Cube, edge $a$	$a^3$	$4a^2$	$6a^2$
Cuboid	$lbh$	$2(l + b)h$	$2(lb + bh + hl)$
Prism	base area $\times h$	base perimeter $\times h$	side surface $+ 2 \times$ base area
Pyramid	$\frac{1}{3} \times$ base area $\times h$	$\frac{1}{2} \times$ base perimeter $\times$ slant height	side surface $+$ base area
Cylinder	$\pi r^2 h$	$2\pi rh$	$2\pi r(r + h)$
Cone	$\frac{1}{3}\pi r^2 h$	$\pi rl$	$\pi r(l + r)$
Sphere	$\frac{4}{3}\pi r^3$	$4\pi r^2$	$4\pi r^2$
Solid hemisphere	$\frac{2}{3}\pi r^3$	$2\pi r^2$	$3\pi r^2$ The total surface includes the flat circle; a bowl has no lid, so it has only the curved part.

In the cone row

l

is the slant height; in the cuboid row

l

is the length.

Watch out for (7)

Adding the percentages
→ How a change in length changes area and volume
Increased by 200% means three times
→ How a change in length changes area and volume
New size or increase
→ How a change in length changes area and volume
Squaring the height too
→ Ratios of two cylinders or two cones
Measuring from the base
→ Slicing a cone by a plane parallel to the base
Height ratio used as the volume ratio
→ Slicing a cone by a plane parallel to the base
Cone against cylinder the wrong way round
→ Volume and surface formulas of the standard solids

Test yourself on Mensuration 3D

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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