Sometimes tested M5.17

Congruence and Similarity

This topic covers how to compare geometric figures. You'll use congruence to identify identical shapes and similarity to relate shapes of different sizes, including how their lengths, areas, and volumes scale.

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

  • Congruent figures are identical in both shape and size; all corresponding angles and side lengths are equal.
  • Similar figures have the same shape but different sizes; corresponding angles are equal, and the ratio of corresponding side lengths is constant.
  • This constant ratio is called the linear scale factor, often denoted 'k'.
  • If the linear scale factor between two similar shapes is 'k', their surface areas scale by a factor of k2.
  • If the linear scale factor is 'k', their volumes (and properties proportional to volume, like mass) scale by a factor of k3.

Formulae

Lengthnew = Lengthold × k

To find a corresponding length on a similar figure when the linear scale factor 'k' is known.

Areanew = Areaold × k2

To find the area of a similar figure when the original area and linear scale factor 'k' are known.

Volumenew = Volumeold × k3

To find the volume of a similar 3D figure when the original volume and linear scale factor 'k' are known.

Definitions

Congruent
Figures that are an exact match. One can be perfectly superimposed on the other through translations, rotations, or reflections.
Similar
Figures that are a perfect enlargement or reduction of each other. They have the same shape, but not necessarily the same size.
Linear Scale Factor (k)
The ratio of a length on one similar figure to the corresponding length on another. k = New Length / Original Length.

Worked example

A small clay pot has a surface area of 100 cm2 and a height of 8 cm. A larger, geometrically similar pot has a height of 24 cm. What is the surface area of the larger pot?

  1. 1

    First, determine the linear scale factor (k) by comparing the heights.

  2. 2
    k = New Height / Old Height = 24 cm / 8 cm = 3
  3. 3

    The question asks for surface area, so we need the area scale factor, which is k2.

  4. 4

    Area scale factor = 32 = 9.

  5. 5

    Finally, multiply the original surface area by the area scale factor to find the new surface area.

  6. 6
    New Area = Old Area × 9 = 100 cm2 × 9 = 900 cm2

Answer: 900 cm2

Common mistakes

  • ×Using the wrong scale factor: Applying the linear scale factor (k) when you should be using the area (k2) or volume (k3) factor. Always check if the question concerns length, area, or volume.
  • ×Arithmetic errors when calculating squares or cubes of the scale factor, especially with fractions or decimals. For example, miscalculating (1.5)2 or (2/3)3.
  • ×Incorrectly identifying corresponding sides on similar triangles, particularly if one triangle is rotated or reflected relative to the other.

No-calculator tips

  • ✓Always simplify the ratio of lengths to find the simplest integer or fractional scale factor before you square or cube it. It's easier to calculate 32 than 242 / 82.
  • ✓Memorise common squares (up to 202) and cubes (up to 103) to speed up calculations for area and volume scale factors.
  • ✓When dealing with fractions, remember that (a/b)n = an / bn. This avoids converting to decimals and simplifies the arithmetic.

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

All Mathematics 1 topics