E. Geometry and Measurement · Geometry and Measurement: Circles, Triangles, Measurement Systems, Pythagoras and Volume

How Changing Dimensions Affects Surface Area and Volume

E1.4 A1About 65 minutes

Learning goals

I can explain why multiplying every dimension of a 3D object by k multiplies its surface area by k² and its volume by k³.
I can predict how the volume of a prism or cylinder changes when only one dimension changes.
I can compare the effect of changing the radius and changing the height of a cylinder.
I can work backwards from a change in volume or surface area to the scale factor.

Words to know

surface area (SA)
The total area of all the faces or surfaces of a 3D object, in square units.

volume (V)
The amount of space a 3D object takes up, in cubic units.

similar solids
3D objects with the same shape, where every length is multiplied by the same scale factor.

cube root
A number that, multiplied by itself three times, gives the original: the cube root of 27 is 3.

square-cube law
When a solid is scaled by k, its surface area grows by k² but its volume grows by k³.

A hook: the giant ice block

An ice sculptor has a block of ice. Another sculptor has the same total amount of ice, but as 64 small cubes. Whose ice melts faster on a sunny February afternoon?

The small cubes melt faster, because together they have much more surface touching the warm air. How surface area and volume change with size explains melting ice, why big animals keep warm more easily, and why a giant ant could not exist.

Remember from Grade 7 and 8

You found the surface area and volume of prisms and cylinders: V = (area of base) × height, and SA = the sum of the areas of all the faces. For a cylinder, V = πr²h and SA = 2πr² + 2πrh.

Cubes built from 1, 8 and 27 unit cubes. When the side is multiplied by k, the surface area is multiplied by k² and the volume by k³.

Scaling a solid by k

If every length of a solid is multiplied by k:

every edge, height, radius and perimeter is multiplied by k,
every face area, and the total surface area, is multiplied by k²,
the volume is multiplied by k³.

This works for every solid: prisms, cylinders, pyramids, cones and spheres.

Side of cube (cm)	Surface area 6s² (cm²)	Volume s³ (cm³)	SA ratio	V ratio
1	6	1	1	1
2	24	8	4	8
3	54	27	9	27
4	96	64	16	64
5	150	125	25	125

Surface area grows by square numbers; volume grows by cube numbers. Build this in a spreadsheet with =6*A2^2 and =A2^3.

Changing only one dimension

For a prism, V = (area of base) × h. Multiply only the height by k and the volume is multiplied by k.
For a cylinder, V = πr²h. Multiply the height by k: V × k. Multiply the radius by k: V × k², because r is squared.
The surface area does not follow a simple rule when only one dimension changes. Recalculate it.

Doubling the height of a cylinder doubles its volume. Doubling the radius makes the base 4 times as big, so the volume is multiplied by 4.

Worked example

Example 1: Scale a cube

A cube has sides of 2 cm. It is enlarged by a scale factor of 3. Find the new surface area and volume.

Original: SA = 6 × 2² = 24 cm², V = 2³ = 8 cm³.
New side = 6 cm. SA = 6 × 6² = 216 cm², V = 6³ = 216 cm³.
Check the ratios: 216 ÷ 24 = 9 = 3², and 216 ÷ 8 = 27 = 3³.

SA = 216 cm² (× 9), V = 216 cm³ (× 27)

Worked example

Example 2: Radius or height?

A cylinder has radius 4 cm and height 10 cm. Compare doubling the height with doubling the radius.

Original: V = π × 4² × 10 = 160π ≈ 502.7 cm³.
Double the height: V = π × 4² × 20 = 320π ≈ 1005.3 cm³ (× 2).
Double the radius: V = π × 8² × 10 = 640π ≈ 2010.6 cm³ (× 4).

Doubling the radius has the bigger effect: × 4 instead of × 2.

Worked example

Example 3: Shrink a box

A box is 10 cm × 20 cm × 30 cm. Every dimension is halved. Find the new surface area and volume.

Original: V = 10 × 20 × 30 = 6000 cm³. SA = 2(10 × 20 + 10 × 30 + 20 × 30) = 2(200 + 300 + 600) = 2200 cm².
k = ½, so SA × ¼ and V × ⅛.
New SA = 2200 ÷ 4 = 550 cm². New V = 6000 ÷ 8 = 750 cm³.
Check: 5 × 10 × 15 = 750. ✓

SA = 550 cm², V = 750 cm³

Worked example

Example 4: Model water tower

A 1 : 20 model of a water tower holds 2.5 L. How much does the real tower hold?

k = 20, so the volume is multiplied by 20³ = 8000.
2.5 L × 8000 = 20 000 L.

20 000 L

Worked example

Example 5: Only the height changes

A box is 2 cm × 3 cm × 4 cm. The 4 cm height is doubled to 8 cm. What happens to the volume and surface area?

V: 2 × 3 × 4 = 24 cm³ becomes 2 × 3 × 8 = 48 cm³ (doubles).
SA: 2(6 + 8 + 12) = 52 cm² becomes 2(6 + 16 + 24) = 92 cm².
92 ÷ 52 ≈ 1.77, so the surface area does not double.

V × 2; SA 52 cm² → 92 cm²

Watch out!

Scale factor k: lengths × k, areas × k², volumes × k³. Check which one the question asks for.
Working backwards: a volume ratio needs a cube root (∛8 = 2, ∛27 = 3, ∛64 = 4). A surface area ratio needs a square root.
In V = πr²h, the radius is squared, so it matters more than the height.

Hands-on activity

Sugar-cube (or building-block) solids

You need: 27 small cubes (sugar cubes, dice or building blocks), paper and a pencil.

Build cubes with sides 1, 2 and 3 units.
For each, count the cubes (volume) and the square faces you can see on all six sides, including the bottom (surface area).
Record your results in a table and compare with 6s² and s³.
Take the 8-cube block apart into 8 single cubes. Count the total surface area now (8 × 6 = 48). Why is it bigger than 24? Connect this to the melting-ice hook.

Model answer: the 2 × 2 × 2 cube has V = 8 and SA = 24 square faces; the 3 × 3 × 3 cube has V = 27 and SA = 54. Splitting the 8 cubes apart exposes the hidden faces, so the SA doubles from 24 to 48.

Just for fun

If an ant were scaled up to the size of a car, its weight would be multiplied by k³ but the strength of its legs (which depends on their cross-section area) only by k². Its legs would buckle. Giant-insect movies are bad math!

Your turn

A cube has sides of 4 cm. Every dimension is doubled. Find the new SA and V.

Two boxes are similar with scale factor 3. The small one has a volume of 50 cm³. Find the large one's volume.

A cylinder has r = 5 cm and h = 8 cm. Find its volume, then the volume if the height is tripled, and if the radius is tripled.

A ball is reduced by a scale factor of ½. By what factor is its volume multiplied?

Two similar tanks have volumes in the ratio 64 : 1. What is the ratio of their heights? Their surface areas?

A solid has SA = 30 cm². It is enlarged by a scale factor of 2. Find the new SA.

Which increases a cylinder's volume more: doubling its radius or doubling its height?

More practice: warm-up

All lengths × 2. Volume × ?

All lengths × 2. Surface area × ?

All lengths × 3. Volume × ?

A 1 cm cube is scaled to a 2 cm cube. Old volume 1 cm³. New volume?

Only the height of a prism is doubled. Volume × ?

More practice: on your own

A can has r = 3.5 cm and h = 12 cm. A similar can is 1.5 times as tall. Find both volumes.

A gift box is a 10 cm cube. It is scaled by 1.2. Find the new SA and V.

A 1 : 10 model car's paint covers 0.04 m². How much surface does the real car's paint cover?

A box is 2 cm × 3 cm × 4 cm. Only the 3 cm side is tripled. Find the new volume and surface area.

Compare the surface area of one 4 cm ice cube with 64 cubes of 1 cm, which have the same total volume.

Challenge

A balloon's radius increases by 10%. By what percent does its volume increase?

You want a similar hot tub that holds twice the water. By what factor must each dimension be multiplied?

A 2 m tall statue needs 3 L of paint and has a mass of 400 kg. A similar statue is 6 m tall. How much paint and what mass?

Check your understanding

Quiz

Every dimension of a box is doubled. Its volume is multiplied by:

Every dimension of a solid is tripled. Its surface area is multiplied by:

A cylinder's radius is doubled and the height is unchanged. Its volume is multiplied by:

Two similar boxes have volumes 10 cm³ and 270 cm³. What is the scale factor?

A prism's height is multiplied by 5 and nothing else changes. Its volume is multiplied by:

Why do 64 small ice cubes melt faster than one big cube of the same volume?

For parents and teachers

Big idea: This lesson continues E1.4 with 3D objects: surface area changes with k² and volume with k³. The table and spreadsheet ideas cover the "using technology" part of the expectation.

Common difficulties:

Using k² for volume. Building cubes with blocks makes k³ visible.
Doubling the radius vs the height of a cylinder. Have your teen point to the squared r in the formula.
Cube roots when working backwards. Learn the first few cubes: 1, 8, 27, 64, 125.

How to help: Compare two similar containers at home (small and large yogurt tubs, cans). Measure the heights, predict the volume ratio, and check the labels.