0580

Congruence and Similarity

Lengths, Areas and Volumes

Powers of the same scale factor

  • Call the enlargement's scale factor k, measured on the lengths. Areas and volumes then scale by powers of that same k
Two similar prisms compared: object A has cross-sectional area 8 cm², length 7 cm and volume 56 cm³, object B has area 32 cm², length 14 cm and volume 448 cm³, so the length scale factor is 2, the area scale factor is 4 and the volume scale factor is 8
Source: Similar areas & volumes by Save My Exams
QuantityScales byRecovering k
Lengthk—
Areak²take the square root
Volumek³take the cube root
  • The powers follow from the units, since an area is two lengths multiplied together and a volume is three
  • Multiplying an area by k instead of k² is the single most common error in this topic
  • Identify first whether each quantity given is a length, an area or a volume, because the whole method turns on that
Worked example

Volume from a length scale factor

Two cones are mathematically similar. The smaller has height 6 cm and volume 90 cm³. The larger has height 9 cm. Find the volume of the larger cone.

Solution:

  • Heights are lengths, so they give the length scale factor
  • k = 9 ÷ 6 = 1.5
  • Volume scales by k³, so the volume scale factor is 1.5³ = 3.375
  • Volume of the larger cone = 90 × 3.375
  • Volume = 303.75 cm³
  • Check as fractions: k = 3/2, so k³ = 27/8, and 90 × 27/8 = 2430/8 = 303.75
Exam tip

"Mathematically similar" is the standard opening, appearing in 15 of the 43 papers. Areas scale by k² and volumes by k³, never by k, and going back from an area or volume to a length needs a square or cube root. Where two similar triangles overlap on the diagram, redraw them separately and facing the same way, or the corresponding sides get paired wrongly.

Worked example

Length from an area scale factor

Two similar shapes have areas of 48 cm² and 108 cm². A side of the smaller shape is 8 cm. Find the corresponding side of the larger.

Solution:

  • The quantities given are areas, so they give the area scale factor
  • Area scale factor = 108 ÷ 48 = 9/4
  • The length scale factor is the square root of this
  • k = √(9/4) = 3/2 = 1.5
  • The missing side is on the larger shape, so multiply
  • 8 × 1.5 = 12 cm
  • Check: the areas would then be in the ratio 1.5² = 2.25, and 108 ÷ 48 = 2.25