Practice Maths

Similarity and Scale

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Key Terms

Similar figures
Shapes that are the same shape but different size; all corresponding angles are equal and corresponding sides are in proportion.
Scale factor k
The ratio of a length in the image to the corresponding length in the original; k = new/original.
Area scale factor
If linear scale factor is k, areas scale by k². (e.g. sides doubled → area quadrupled.)
Volume scale factor
If linear scale factor is k, volumes scale by k³.
Map scale
A ratio such as 1 : 50 000 means 1 cm on the map represents 50 000 cm (500 m) in reality.
Enlargement / reduction
k > 1 produces an enlargement; 0 < k < 1 produces a reduction.

Similar figures have the same shape but different sizes. Corresponding angles are equal and corresponding sides are in the same ratio. This ratio is called the scale factor, denoted k.

Scale Factor Rules

MeasurementRatioRelationship
Lengths (sides, perimeter)kLarge = k × small
Areas (surface area)k²Large area = k² × small area
Volumesk³Large volume = k³ × small volume

The scale factor k must be the ratio of corresponding lengths: k = (length in large shape) ÷ (corresponding length in small shape). If k > 1 the shape is enlarged; if k < 1 it is reduced.

Map scales: A scale of 1:n means 1 unit on the map = n units in reality. Actual distance = map distance × n.

Small triangle a b c × k Large triangle ka kb kc Lengths: ratio k Areas: ratio k²   Volumes: ratio k³

Worked Example 1 — Finding the scale factor

Two similar rectangles: small is 3 cm × 5 cm, large is 9 cm × 15 cm. Find the scale factor and the ratio of areas.

k = 9 ÷ 3 = 3 (verify: 15 ÷ 5 = 3 ✓)

Area of small = 3 × 5 = 15 cm²

Area of large = 9 × 15 = 135 cm²

Ratio of areas = 135 ÷ 15 = 9 = k² = 3² ✓

Scale factor k = 3, area ratio = 9.

Worked Example 2 — Volume ratio

Two similar cylinders have radii 4 cm and 10 cm. The small has volume 100π cm³. Find the volume of the large.

k = 10 ÷ 4 = 2.5

Volume ratio = k³ = 2.5³ = 15.625

Large volume = 15.625 × 100π = 1562.5π ≈ 4908.74 cm³

Hot Tip: Always identify the scale factor from lengths first. Then square it for area comparisons and cube it for volume comparisons. Never apply the k² or k³ multiplier to lengths, or the k multiplier to areas — match the power to the dimension.

Full Lesson

Similarity is one of the most powerful ideas in measurement. Once you know two shapes are similar, you can find any unknown measurement using just the scale factor. This applies to maps, scale models, photography enlargements, and engineering drawings.

What Makes Shapes Similar?

Two shapes are similar if:

  • All corresponding angles are equal, AND
  • All corresponding side lengths are in the same ratio (the scale factor k).

For triangles, you only need to show angles are equal (AA test) or sides are in ratio (SSS test) to prove similarity.

Calculating the Scale Factor

Given two similar figures, k = (measurement in new figure) ÷ (corresponding measurement in original). Always use the same type of measurement (e.g. length to length, not length to area).

Map Scales

A map scale of 1 : 50 000 means 1 cm on the map = 50 000 cm = 500 m in reality. To find actual distance: multiply the map measurement by the scale factor (50 000). To find map distance: divide the actual distance by the scale factor.

Guide Example — Map scale calculation

A map has scale 1 : 250 000. Two towns are 8.4 cm apart on the map. Find the actual distance in km.

Actual distance = 8.4 × 250 000 cm = 2 100 000 cm

Convert to km: 2 100 000 ÷ 100 000 = 21 km

Guide Example — Scale factor and area

A photo is 4 cm × 6 cm. It is enlarged by scale factor 3.

New dimensions: 12 cm × 18 cm

Original area = 24 cm²; New area = 216 cm²

Area ratio = 216 ÷ 24 = 9 = 3² = k² ✓

Tip: When working with volumes and scale factors, always find k from lengths first. If given volumes, work backwards: k³ = Vlarge ÷ Vsmall, then k = ³√(k³).

Mastery Practice

See Answers ➔

  1. Q1 — Scale factor of similar rectangles

    Fluency

    Two similar rectangles: the small one has dimensions 4 cm × 6 cm; the large one has dimensions 10 cm × 15 cm. Find the scale factor.

  2. Q2 — Using scale factor to find lengths and areas

    Fluency

    The scale factor from original to enlarged is k = 3. The original has a side of length 8 cm and area 24 cm².

    (a) Find the corresponding side length in the enlarged figure.

    (b) Find the area of the enlarged figure.

  3. Q3 — Map scale distance

    Fluency

    A map has scale 1 : 50 000. A road measures 6.4 cm on the map. Find the actual distance in km.

  4. Q4 — Similar triangles: scale factor and side lengths

    Understanding

    Two similar triangles: the small triangle has sides 5 cm, 7 cm, and 9 cm. The large triangle has a perimeter of 63 cm.

    (a) Find the scale factor.

    (b) Find each side of the large triangle.

  5. Q5 — Scale factor and volume

    Understanding

    The scale factor (original to enlarged) is k = 2.5. The original solid has volume 48 cm³. Find the volume of the enlarged solid.

  6. Q6 — Photo enlargement

    Understanding

    A photo is 10 cm × 15 cm. It is enlarged so the longer side becomes 24 cm.

    (a) Find the scale factor.

    (b) Find the new shorter side.

    (c) Find the ratio of the enlarged area to the original area.

  7. Q7 — Similar cylinders: dimensions and volume

    Understanding

    Two similar cylinders: the small has r = 4 cm and h = 6 cm. The scale factor (small to large) is k = 1.5.

    (a) Find the dimensions of the large cylinder.

    (b) Find the ratio of surface areas.

    (c) Find the ratio of volumes.

    (d) Find the volume of the large cylinder.

  8. Q8 — Model car at scale 1:24

    Understanding

    A model car is built at scale 1 : 24. The real car is 4.56 m long, 1.8 m wide, and 1.44 m tall.

    (a) Find the model dimensions in cm.

    (b) Find the ratio of the model’s surface area to the real car’s surface area.

    (c) If the model’s fuel tank holds 12 mL, find the real tank capacity in litres.

  9. Q9 — Similar pyramids: finding scale factor from volumes

    Problem Solving

    Two similar pyramids have volumes 64 cm³ and 216 cm³.

    (a) Find the ratio of their volumes (in simplest form).

    (b) Find the scale factor (length ratio).

    (c) Find the ratio of their surface areas.

    (d) The small pyramid has a square base with side 4 cm. Find the base side of the large pyramid.

  10. Q10 — Similar spherical tanks and painting cost

    Problem Solving

    A factory makes spherical tanks in two sizes. The large tank has radius 1.5 m. The small tank is geometrically similar with a scale factor of 2 : 3 (small : large).

    (a) Find the radius of the small tank.

    (b) Find the surface area of each tank.

    (c) Find the volume of each tank.

    (d) If the large tank costs $8100 to paint, find the cost to paint the small tank (assuming the same cost per m²).