Similarity
Section: Geometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Similar Figures and Scale Factor
Two shapes are similar if they are the same shape but not necessarily the same size - one is an enlargement of the other. In similar figures, all corresponding sides are in the same ratio, called the scale factor.
- Corresponding sides of similar figures are always in the same ratio (the scale factor)
- To find the scale factor, divide a length on one shape by the corresponding length on the other
- To find an unknown length, multiply (or divide) the known corresponding length by the scale factor
- Corresponding angles in similar figures are always equal
Worked Example: Finding an Unknown Length Using Similar Triangles
- Question: Triangles ABC and PQR are similar, with AB corresponding to PQ, and AC corresponding to PR. AB = 6 cm, PQ = 15 cm, AC = 8 cm. Find PR.
- Step 1: Scale factor: PQ÷ AB=156=2.5
- Step 2: PR=AC2.5=82.5=20
- Answer: PR = 20 cm
Common Mistakes
MistakeDividing the lengths the wrong way round, giving the reciprocal of the intended scale factor
Fixalways divide the new (target) shape's length by the corresponding original shape's length to get the scale factor multiplying from original to new
MistakeMatching sides that are not actually corresponding, e.g. comparing the longest side of one triangle to the shortest of the other
Fixidentify corresponding sides by the vertices they connect, matching the order the shapes are named in (e.g. AB corresponds to PQ if triangle ABC ~ triangle PQR)
Area of Similar Figures
Extended Only
When two figures are similar, their areas are not in the same ratio as their lengths - the area ratio is the square of the length (scale) ratio.
- If the length scale factor between two similar figures is k, the area scale factor is k^2
- To find an unknown area, multiply the known area by the square of the length scale factor
- To find a length scale factor from a given area ratio, take the square root of the area ratio
Worked Example: Finding an Area Using the Length Scale Factor
- Question: Two similar rectangles have corresponding sides in the ratio 2:5. The smaller rectangle has an area of 12 cm². Find the area of the larger rectangle.
- Step 1: Length scale factor: (5/2)
- Step 2: Area scale factor: ((5/2))^2=(25/4)
- Step 3: Area of larger rectangle: 12×(25/4)=75
- Answer: 75 cm²
Common Mistakes
MistakeUsing the length scale factor directly for an area calculation, without squaring it
Fixarea always scales by the square of the length scale factor - square the ratio before multiplying
MistakeSquaring the areas themselves, instead of squaring the length scale factor
Fixonly the length ratio is squared to get the area ratio - the areas are multiplied by this squared ratio, not squared themselves
Volume of Similar Solids
Extended Only
For similar solids, the volume ratio is the cube of the length (scale) ratio - matching the same pattern as areas, but with a power of 3 instead of 2.
- If the length scale factor between two similar solids is k, the volume scale factor is k^3
- To find an unknown volume, multiply the known volume by the cube of the length scale factor
- To find a length scale factor from a given volume ratio, take the cube root of the volume ratio
Worked Example: Finding a Length from a Volume Ratio
- Question: Two mathematically similar containers have capacities of 2.16 litres and 0.64 litres. The larger container has a height of 18 cm. Find the height of the smaller container.
- Step 1: Volume ratio: 2.160.64=3.375
- Step 2: Length scale factor: √[3]3.375=1.5
- Step 3: Height of smaller container: 181.5=12
- Answer: 12 cm
Common Mistakes
MistakeTaking the square root instead of the cube root when converting a volume ratio into a length scale factor
Fixvolumes scale by the cube (power 3) of the length ratio, so a cube root (not a square root) is needed to reverse this
MistakeMultiplying by the length scale factor directly instead of the cube of it, when finding an unknown volume from a known one
Fixfor volume calculations, the known volume must be multiplied by the length scale factor cubed, not by the scale factor itself
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