CDS Mathematics · Mensuration 3D

Scaling & Comparing Solids

How volume and surface change when a solid is scaled, how the ratio of two cones or cylinders follows from their radii and heights, and what equal volume or equal surface forces.

All Mensuration 3D notes CDS Mathematics strategy

Why this matters

Easy marks when the rule is clear, and the most common source of wrong answers when it is not. A third of the page is HARD, all of it in the equal-surface and equal-volume comparisons.

Concept 1 of 3: Similar solids: lengths, squares, cubes

Scale every length of a solid by

k

: surfaces grow by

k^2

, volumes by

k^3

. Go the other way with roots — a volume ratio needs a cube root to become a length ratio.

Definition

Lengths $\times k$ ⇒ surface $\times k^2$ , volume $\times k^3$ .
Surface ratio $a : b$ ⇒ length ratio $\sqrt a : \sqrt b$ ⇒ volume ratio $a^{3/2} : b^{3/2}$ .
A $p\%$ increase in surface ⇒ lengths $\times\sqrt{1 + \tfrac{p}{100}}$ .
Different stretches on each edge (cuboid): multiply the factors.

Scale factor k

\frac{S_1}{S_2} = k^2, \qquad \frac{V_1}{V_2} = k^3

Worked example

The volume of a sphere is increased by

237.5\%

. By what percentage does its surface area increase?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (II) 2019 — Elementary Mathematics · Q73Moderate

Example 1 · Mensuration 3D · Scaling and Comparing Solids

The volume of a spherical balloon is increased by 700%. What is the percentage increase in its surface area ?

An increase of p% is a factor of 1 + p/100

300\%

more is

4

times, not

3

. Convert the percentage to a factor before taking roots.

Drill 4 more on similar solids: lengths, squares, cubes

Concept 2 of 3: Ratios of cones and cylinders

For both cones and cylinders the volume is a constant times

r^2h

. So a ratio question never needs

\pi

or the

\dfrac13

: compare

r^2h

directly.

Definition

$\dfrac{V_1}{V_2} = \dfrac{r_1^2 h_1}{r_2^2 h_2}$ for two cones, or two cylinders.
Equal volumes: $\dfrac{r_1^2}{r_2^2} = \dfrac{h_2}{h_1}$ .
Radius up $p\%$ at the same height: volume up $\left(1 + \tfrac{p}{100}\right)^2 - 1$ , i.e. $p\left(2 + \tfrac{p}{100}\right)\%$ .

Cones or cylinders

\frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2}

Worked example

Two cylinders have radii in the ratio

2 : 3

and heights in the ratio

5 : 4

. Find the ratio of their volumes.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2020 · CDS (II) 2020 — Elementary Mathematics · Q77Moderate

Example 2 · Mensuration 3D · Scaling and Comparing Solids

The volumes of two cones are in the ratio 1 : 4 and their diameters are in the ratio 4 : 5. What is the ratio of their heights ?

Drill 4 more on ratios of cones and cylinders

Concept 3 of 3: Equal volume or equal surface

Set the two expressions equal and read off the relation. The facts worth knowing by heart: a cone and a hemisphere on the same base have equal volume when the cone is twice as tall; for a given surface the sphere holds the most.

Definition

Cone and hemisphere, same base, equal volume: $h = 2r$ .
Cone, hemisphere and cylinder of radius and height $r$ : volumes $1 : 2 : 3$ .
Cube and sphere with equal surface: the sphere has the larger volume; $x^2 : y^2 = \pi : 6$ for cube : sphere volumes.
A cuboid recast into a cube has LESS surface: the cube minimises surface among cuboids of a given volume.

Cone : hemisphere : cylinder (radius r, height r)

\tfrac13\pi r^3 : \tfrac23\pi r^3 : \pi r^3 = 1 : 2 : 3

Worked example

A cone, a hemisphere and a cylinder stand on equal bases of radius

r

with equal heights. Their total volume equals that of a sphere of radius

R

. Find

R

in terms of

r

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q61Easy

Example 3 · Mensuration 3D · Scaling and Comparing Solids

A cone and a hemisphere have equal bases and equal volumes. What is the ratio of the height of the cone to the radius of the hemisphere?

Drill 8 more on equal volume or equal surface

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)

Similar solids: lengths, squares, cubes
Scale factor k
$\frac{S_1}{S_2} = k^2, \qquad \frac{V_1}{V_2} = k^3$
Ratios of cones and cylinders
Cones or cylinders
$\frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2}$
Equal volume or equal surface
Cone : hemisphere : cylinder (radius r, height r)
$\tfrac13\pi r^3 : \tfrac23\pi r^3 : \pi r^3 = 1 : 2 : 3$

Watch out for (1)

An increase of p% is a factor of 1 + p/100
→ Similar solids: lengths, squares, cubes

Test yourself on Mensuration 3D

20 past CDS questions from this chapter, timed at 24 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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