Year 11 General Shape And Measurement

Similar Solids

5 practice questions 2 video lessons Theory + worked examples

Theory

Two solids are similar when one is an enlargement of the other — same shape, different size. This page covers the linear, surface area and volume scaling rules in 3D, with worked examples on tank capacities, painting costs, and going backwards from area or volume ratios to the linear ratio.

Two solids are similar when one is an enlargement of the other: same shape but a different size, with all matching lengths in the same ratio. Cubes are always similar to other cubes; spheres to other spheres; pairs of cylinders, cones and pyramids are similar when their proportions match.

The fundamental rule extends the 2D similarity result into three dimensions. If two similar solids have linear scale ratio $a : b$ , then surface areas scale as $a^{2} : b^{2}$ and volumes scale as $a^{3} : b^{3}$ . Volume picks up the extra factor because there are three independent dimensions to scale.

You can also go the other way: from a surface area ratio, take the square root to get the linear ratio; from a volume ratio, take the cube root. For example, a volume ratio of $27 : 64$ gives a linear ratio of $\sqrt[3]{27} : \sqrt[3]{64} = 3 : 4$ .

Two similar cubes. Linear ratio

1 : 2

, surface area ratio

1 : 4

, volume ratio

1 : 8

Two similar cones with matching proportions and linear ratio

1 : 2

If two similar solids have linear scale ratio $a : b$ :

Quantity	Ratio
Lengths (sides, radii, heights)	$a : b$
Surface areas	$a^{2} : b^{2}$
Volumes (and capacities)	$a^{3} : b^{3}$

Going backwards — from an area or volume ratio to the linear ratio:

linear ratio = \sqrt{surface area ratio}

linear = \sqrt{SA ratio}

linear ratio = \sqrt[3]{volume ratio}

linear = \sqrt[3]{V ratio}

Real-world rule of thumb. Costs that scale with surface area (paint, plating, wrapping) use $k^{2}$ . Costs that scale with volume (concrete, water, sand) use $k^{3}$ . Pick the right power for the situation.

How to solve any similar-solids problem

Identify the linear ratio $a : b$ . It may be given directly, or you may need to derive it from a length, area, or volume.
Decide whether you need the area ratio ( $a^{2} : b^{2}$ ) or volume ratio ( $a^{3} : b^{3}$ ) based on the quantity asked for.
Set up a proportion using matching quantities and solve for the unknown.

Astronomy and biology examples. A planet $2$ times the diameter of another has $8$ times the volume. A baby that doubles in linear size has $4$ times the skin and $8$ times the body mass.

Example 1 — Volume from Linear Ratio

Two similar tanks have a linear ratio of

2 : 3

. The smaller tank holds

80

L. Find the capacity of the larger.

Solution

Volume ratio is $2^{3} : 3^{3} = 8 : 27$ .

$\frac{80}{V_{large}}$	$=$	$\frac{8}{27}$
$V_{large}$	$=$	$\frac{80 \times 27}{8}$
$V_{large}$	$=$	$270 L$

V_{large} = 270 L

Example 2 — Linear from Volume

Two similar solids have volumes

64

^{3}

and

729

^{3}

. Find the linear ratio of the smaller to the larger.

Solution

Take the cube root of the volume ratio.

$\frac{a}{b}$	$=$	$\sqrt[3]{\frac{64}{729}}$
$\frac{a}{b}$	$=$	$\frac{4}{9}$

Linear ratio is $4 : 9$ .

a : b = 4 : 9

Example 3 — Linear from Surface Area

Two similar statues have surface areas of

36

^{2}

and

100

^{2}

. Find the linear ratio.

Solution

Take the square root of the area ratio.

$\frac{a}{b}$	$=$	$\sqrt{\frac{36}{100}}$
$\frac{a}{b}$	$=$	$\frac{6}{10} = \frac{3}{5}$

Linear ratio is $3 : 5$ .

a : b = 3 : 5

Example 4 — Painting Cost

A small statue costs

$ 60

to paint. A larger similar statue has a linear ratio of

1 : 3

compared to the smaller. Find the painting cost of the larger statue.

Solution

Painting scales with surface area, so use the square of the linear ratio.

area ratio	$=$	$1^{2} : 3^{2} = 1 : 9$
cost	$=$	$60 \times 9$
cost	$=$	$$ 540$

cost = 540

Common pitfalls

Square-rooting a ratio incorrectly.

\sqrt{4 : 9} = 2 : 3

, not

2 : 4.5

. Take the root of each number separately.

Wrong power for the quantity. Costs that scale with surface area (paint, plating, wrapping) use

k^{2}

. Costs that scale with volume (concrete, water, sand) use

k^{3}

. Pick the right one.

Cube-rooting volume ratios. Going from a volume ratio to a linear ratio means taking the cube root, not the square root. Volume ratio

8 : 125

gives linear ratio

2 : 5

Mixed units in volumes. Always convert capacities to the same unit before substituting (e.g.

1

^{3}

= 1000

L). Otherwise the ratio is meaningless.

Frequently asked questions

What are similar solids?

Two solids are similar when one is an enlargement of the other — same shape but a different size, with all matching lengths in the same ratio. Cubes are always similar to other cubes, spheres to other spheres, and pairs of cylinders, cones, etc. are similar when their proportions match.

If two similar solids have linear ratio a to b, what are the area and volume ratios?

Surface areas scale as a squared to b squared. Volumes and capacities scale as a cubed to b cubed. So solids with linear ratio 1 to 3 have surface area ratio 1 to 9 and volume ratio 1 to 27.

How do I find the linear ratio if I only know the volume ratio?

Take the cube root of each part of the volume ratio. For example, volume ratio 27 to 64 gives linear ratio cube root of 27 to cube root of 64, which is 3 to 4.

How do I find the linear ratio if I only know the surface area ratio?

Take the square root of each part. For example, surface area ratio 36 to 100 gives linear ratio square root of 36 to square root of 100, which is 6 to 10 or 3 to 5.

Why does painting cost scale with the square of the linear ratio?

Because paint covers area. If you double the linear dimensions of an object, the surface area is multiplied by 2 squared which is 4, so you need 4 times as much paint and pay 4 times as much.

Why does volume scale faster than surface area?

Because volume depends on three dimensions (length, width, height) but surface area depends on only two. Scaling each linear dimension by k multiplies surface area by k squared and volume by k cubed.