Most tested M3.10

Area and Volume Scale Factors

This topic covers the relationship between the dimensions of mathematically similar shapes. Mastering how to scale lengths, areas, and volumes using ratios and scale factors is crucial for solving geometry problems quickly.

Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • If two shapes are similar, their corresponding angles are equal and the ratio of their corresponding lengths is constant.
  • The scale factor for area is the square of the length scale factor (k2), and the volume scale factor is the cube of the length scale factor (k3).
  • Trigonometric ratios (sin, cos, tan) are identical for corresponding angles in similar triangles because they are ratios of side lengths, so the scale factor cancels out.
  • To get from a volume ratio to an area ratio (or vice versa), you must use the length ratio as an intermediate step. Do not convert directly.
  • A ratio of lengths a:b implies a ratio of areas a2:b2 and a ratio of volumes a3:b3.

Formulae

Length Ratio (a:b) → Area Ratio (a2:b2) → Volume Ratio (a3:b3)

Use this core relationship to convert between ratios of different dimensions for any two similar shapes. Remember to always work via the length ratio when moving between area and volume.

Definitions

Similar Shapes
Two shapes are similar if one is an enlargement of the other. This means all corresponding angles are identical, and the ratio between all corresponding lengths is constant.
Linear Scale Factor (k)
The constant multiplier that relates the length of a side on one similar shape to the corresponding side on another. New length = k × Original length.

Worked example

Two solid metal statues, P and Q, are mathematically similar. The ratio of the surface area of P to the surface area of Q is 4:9. The mass of statue Q is 54 kg. Given that mass is proportional to volume, what is the mass of statue P?

  1. 1

    Start with the given Area Ratio:

    Ap :

    Aq = 4 :

    9.

  2. 2

    Find the Length Ratio by taking the square root:

    Lp :

    Lq = √(4) :
    √(9) = 2 :

    3.

  3. 3

    Find the Volume Ratio by cubing the Length Ratio:

    Vp :

    Vq = 23 :
    33 = 8 :

    27.

  4. 4

    Since mass is proportional to volume, the Mass Ratio is the same as the Volume Ratio:

    Mp :

    Mq = 8 :

    27.

  5. 5

    Set up a proportion to find the mass of P:

    Mp / Mq = 8 / 27

    So, Mp / 54 = 8 / 27.

  6. 6

    Solve for Mp:

    Mp = (8 / 27) × 54

    Since 54 / 27 = 2, Mp = 8 × 2 = 16 kg.

Answer: 16 kg

Common mistakes

  • ×Using the wrong power: Applying the length scale factor 'k' directly to an area or volume instead of using 'k2' or 'k3'. For example, if lengths are tripled, the area is 32=9 times larger, not 3 times.
  • ×Skipping the length ratio: Attempting to convert directly between an area ratio and a volume ratio. You MUST find the length ratio first by square-rooting the area ratio or cube-rooting the volume ratio.
  • ×Arithmetic errors with powers: Miscalculating squares and cubes under pressure, for example, thinking 43 is 12 instead of 64. This is a frequent source of incorrect answers.

No-calculator tips

  • ✓Memorise common cubes: Know at least 13 to 63 (1, 8, 27, 64, 125, 216) and their corresponding roots. This makes converting from volume ratios to length ratios instant.
  • ✓Simplify ratios early: Before taking roots, simplify the ratio completely. It is far easier to find the cube root of 1:8 than the cube root of 10:80.
  • ✓Factorise to find roots: If faced with a large number, use prime factorisation. For example, to find cbrt(1728), you might notice it's even: 1728 = 8 × 216. cbrt(1728) = cbrt(8) × cbrt(216) = 2 × 6 = 12.

Read this topic in the official UAT-UK ESAT guide →

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