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Geometry · Topic 6 of 8

Similarity

Video lesson3 worked examples

Theory

Similarity involves the interrelationship of scale across length, area, and volume.

  • Scale Factor (Length): Scale Factor=Corresponding Small SideLarge Side\text{Scale Factor} = \frac{\text{Corresponding Small Side}}{\text{Large Side}} (or vice versa depending on the direction of scaling).
  • Area Factor: The Area Factor is equal to the (Scale Factor)2(\text{Scale Factor})^2.
  • Volume Factor: The Volume Factor is equal to the (Scale Factor)3(\text{Scale Factor})^3.

The Golden Rule: find the linear scale factor first, as new lengthold length\frac{\text{new length}}{\text{old length}}. For lengths, multiply by the scale factor; for areas multiply by its square; for volumes multiply by its cube.

⚠️ Common Examiner Traps

  • Using the linear factor for area/volume: areas scale by k2k^2 and volumes by k3k^3, not by kk.
  • Scaling the wrong way: to go from small to large multiply (factor > 1); from large to small the factor is less than 1.
  • Matching corresponding sides: the scale factor must use a pair of corresponding sides — line the two shapes up the same way round.
  • Proving similar: shapes are similar only if every pair of corresponding sides shares the same ratio.

Worked examples

Example 1

Similar Triangles (lengths)

Triangles ABC and DEF are mathematically similar. Side AB = 6 cm corresponds to DE = 9 cm, and BC = 8 cm. Calculate the length of the corresponding side EF.

Step 1: Find the linear scale factor from the matching pair: DEAB=96=1.5\frac{DE}{AB} = \frac{9}{6} = 1.5.

Step 2: EF corresponds to BC, so multiply by the scale factor: EF=8×1.5EF = 8 \times 1.5.

Answer: EF = 12 cm.

Example 2

Area Similarity

Two similar badges are made. The small badge has a width of 3 cm and an area of 12 cm2. The large badge has a width of 9 cm. Calculate the area of the large badge.

Step 1: Find the length Scale Factor (SF): 9 / 3 = 3.

Step 2: Find the Area Factor: SF2 = 32 = 9.

Step 3: Multiply the small area by the Area Factor: 12 × 9.

Answer: 108 cm2.

Example 3

Volume Similarity

Two mathematically similar coffee cups exist. The large cup has a height of 15 cm and holds 500 ml. The small cup has a height of 12 cm. Calculate the volume of the small cup.

Step 1: Find the length reduction SF: 12 / 15 = 0.8.

Step 2: Find the Volume Factor: 0.83 = 0.512.

Step 3: Multiply the large volume by the Volume Factor: 500 × 0.512.

Answer: 256 ml.

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11 Similarity questions with answers and video solutions