IGCSE Maths · Geometry · Core & Extended

Similarity and Symmetry: Scaling Shapes and Finding Balance

Similarity is about shapes that are the same but scaled; symmetry is about a single shape balancing against itself. Calculating lengths of similar shapes, line symmetry, and rotational symmetry are shared by both tiers. Area/volume scale factors, proving similarity, and 3D symmetry are Extended-only, tagged throughout.

1. Similar Shapes and Length Scale Factor

Similar shapes have exactly the same angles and proportions, just at a different size. The scale factor is the number you multiply every length by to get from one shape to the other.

Similar triangles 4 cm 8 cm The larger triangle is an exact scaled-up copy of the smaller one — same angles, sides in the same ratio.
Worked example Two similar triangles have corresponding bases of 4 cm and 10 cm. The smaller triangle’s height is 3 cm. Find the larger triangle’s height.
Scale factor = 10 ÷ 4 = 2.5
Larger height = 3 × 2.5 = 7.5 cm
Try it — Length Scale Factor
  1. Two similar rectangles have widths 6 cm and 15 cm. The smaller one has height 4 cm. Find the larger rectangle’s height.
  2. Two similar shapes have a scale factor of 3. A side on the smaller shape is 5 cm. Find the corresponding side on the larger shape.
  3. Two similar shapes have corresponding sides 8 cm and 20 cm. A different side on the larger shape is 15 cm. Find the corresponding side on the smaller shape.
Show answers
  1. Scale factor = 15÷6 = 2.5. Height = 4×2.5 = 10 cm
  2. 5×3 = 15 cm
  3. Scale factor = 20÷8 = 2.5. Smaller side = 15÷2.5 = 6 cm

2. Area Scale Factor Extended only

When lengths scale by a factor k, area scales by k² — because area is a length multiplied by a length.

Area of A / Area of B = (Length of A / Length of B)²
Worked example Two similar shapes have a length scale factor of 2.5. The smaller shape has area 8 cm². Find the larger shape’s area.
Area scale factor = 2.5² = 6.25
Larger area = 8 × 6.25 = 50 cm²
Worked example — working backwards from areas Two similar shapes have areas 18 cm² and 200 cm². Find the length scale factor.
Area scale factor = 200 ÷ 18 = 100/9
Length scale factor = √(100/9) = 10/3
Try it — Area Scale Factor Extended
  1. Two similar shapes have a length scale factor of 3. The smaller shape has area 12 cm². Find the larger area.
  2. Two similar shapes have areas 20 cm² and 45 cm². Find the length scale factor.
  3. Two similar triangles have a length scale factor of 4. If the larger triangle has area 240 cm², find the smaller triangle’s area.
Show answers
  1. 3²=9. 12×9 = 108 cm²
  2. 45÷20=2.25. √2.25 = 1.5
  3. 4²=16. 240÷16 = 15 cm²

3. Volume Scale Factor Extended only

Volume involves three lengths multiplied together, so it scales by k³.

Volume of A / Volume of B = (Length of A / Length of B)³
Worked example Two similar solids have a length scale factor of 2.5. The smaller solid has volume 20 cm³. Find the larger solid’s volume.
Volume scale factor = 2.5³ = 15.625
Larger volume = 20 × 15.625 = 312.5 cm³
Worked example — working backwards from volumes Two similar solids have volumes 27 cm³ and 64 cm³. Find the length scale factor.
Volume scale factor = 64/27
Length scale factor = ³√(64/27) = 4/3
Try it — Volume Scale Factor Extended
  1. Two similar solids have a length scale factor of 2. The smaller solid has volume 15 cm³. Find the larger volume.
  2. Two similar solids have volumes 8 cm³ and 125 cm³. Find the length scale factor.
  3. Two similar bottles have heights 10 cm and 25 cm. The smaller bottle holds 200 ml. Find the capacity of the larger bottle.
Show answers
  1. 2³=8. 15×8 = 120 cm³
  2. 125÷8=15.625. ³√15.625 = 2.5
  3. Length scale factor = 25÷10 = 2.5. Volume scale factor = 2.5³ = 15.625. Capacity = 200×15.625 = 3125 ml

4. Proving Two Triangles Are Similar Extended only

Two triangles are similar if their angles match — this is usually shown using parallel lines (which create equal corresponding or alternate angles) plus any shared angle.

Worked example A D E B C In triangle ABC, D lies on AB and E lies on AC, with DE parallel to BC. Show that triangles ADE and ABC are similar.
• Angle A is shared by both triangles (it’s the same angle).
• Angle ADE = angle ABC (corresponding angles, since DE is parallel to BC).
• Angle AED = angle ACB (corresponding angles, since DE is parallel to BC).
All three angles match, so triangle ADE is similar to triangle ABC (AA — two matching angles is enough, since angles in a triangle always sum to 180°, making the third automatically equal too).
Try it — Proving Similarity Extended
  1. If two triangles have two pairs of equal angles, why must the third pair also be equal?
  2. In the diagram above, if AD = 3 cm, AB = 9 cm, and DE = 4 cm, find BC.
Show answers
  1. Angles in a triangle always sum to 180°; if two pairs already match, the remaining angle in each triangle must be 180° minus the same two values, so it’s forced to match too.
  2. Scale factor = AB/AD = 9/3 = 3. BC = DE × 3 = 4 × 3 = 12 cm

5. Line Symmetry

A shape has a line of symmetry if it can be folded along that line so both halves match exactly.

Examples Square: 4 lines Isosceles: 1 line
ShapeLines of symmetry
Equilateral triangle3
Isosceles triangle1
Square4
Rectangle (non-square)2
Rhombus2
Regular pentagon5
Regular hexagon6
Try it — Line Symmetry
  1. How many lines of symmetry does a regular octagon have?
  2. How many lines of symmetry does a scalene triangle have?
  3. How many lines of symmetry does a parallelogram (non-rhombus, non-rectangle) have?
Show answers
  1. 8 (a regular polygon with n sides always has n lines of symmetry)
  2. 0 — no two sides or angles match, so no fold line works.
  3. 0 — despite looking balanced, a general parallelogram has no line of symmetry (only rotational symmetry).

6. Rotational Symmetry

The order of rotational symmetry is how many times a shape looks exactly the same during one full 360° turn.

ShapeOrder of rotational symmetry
Equilateral triangle3
Square4
Rectangle (non-square)2
Parallelogram2
Regular pentagon5
Isosceles triangle (not equilateral)1 (no rotational symmetry beyond the trivial full turn)
Try it — Rotational Symmetry
  1. What is the order of rotational symmetry of a regular hexagon?
  2. What is the order of rotational symmetry of a kite?
  3. A shape has rotational symmetry of order 5. Through what angle does it rotate to look the same each time?
Show answers
  1. 6
  2. 1 (a general kite has no rotational symmetry beyond the full 360° turn)
  3. 360 ÷ 5 = 72°

7. Symmetry Properties of Solids Extended only

In three dimensions, symmetry is described using planes of symmetry (a flat “mirror” slicing the solid into two identical halves) and axes of symmetry (a line the solid can rotate around and look unchanged).

SolidPlanes of symmetry
Cube9
Cuboid (non-cube)3
CylinderInfinitely many through the axis, plus 1 perpendicular to the axis through its midpoint
ConeInfinitely many, all containing the axis
Square-based pyramid (apex directly above centre)4
Try it — 3D Symmetry Extended
  1. How many planes of symmetry does a cuboid (with three different edge lengths) have?
  2. Why does a cylinder have infinitely many planes of symmetry?
  3. How many planes of symmetry does a cube have?
Show answers
  1. 3
  2. Any plane that contains the central axis slices it into two identical mirror-image halves — and there are infinitely many such planes, since the axis can be sliced at any angle around its length.
  3. 9
Common mistakes to watch for
  • Using the length scale factor directly for an area or volume question — remember to square it for area, cube it for volume.
  • Forgetting to take a square root (for area) or cube root (for volume) when working backwards from a scale factor.
  • Assuming every quadrilateral with “matching” sides has line symmetry — a general parallelogram has rotational symmetry but no line symmetry at all.
  • Confusing order of rotational symmetry with the number of lines of symmetry — they’re related for regular polygons but are genuinely different properties.
  • Forgetting the “trivial” full 360° turn always counts, but a shape is only said to have rotational symmetry if it matches at some smaller angle too (order greater than 1).

8. Quick-Fire Challenge Round

True or False?
  1. If the length scale factor is 3, the area scale factor is 9.
  2. If the length scale factor is 3, the volume scale factor is 9.
  3. A regular polygon with n sides has n lines of symmetry.
  4. Every shape with rotational symmetry also has line symmetry.
Show answers
  1. True — 3² = 9. Ext.
  2. False — volume scale factor is 3³ = 27, not 9. Ext.
  3. True.
  4. False — a parallelogram has rotational symmetry (order 2) but no line symmetry at all.
Spot the Error
  1. “Two similar shapes have length scale factor 4, so their areas are also 4 times bigger.”
  2. “Two similar solids have volumes 8 cm³ and 27 cm³, so the length scale factor is 27/8 = 3.375.”
  3. “A rectangle has 4 lines of symmetry, just like a square.”
Show answers
  1. Area scales by the square of the length factor: 4² = 16 times bigger, not 4 times.
  2. The volume ratio itself is 27/8, but the length scale factor is the cube root of that: ³√(27/8) = 3/2, not 3.375.
  3. A non-square rectangle has only 2 lines of symmetry (horizontal and vertical through the middle), not 4 — the two diagonals don’t work as fold lines unless it’s a square.

9. Mixed Practice — Bringing It All Together

Core & Extended
  1. Two similar shapes have corresponding sides 5 cm and 12.5 cm. Find the scale factor.
  2. How many lines of symmetry does a regular pentagon have?
  3. What is the order of rotational symmetry of a square?
Show answers
  1. 12.5÷5 = 2.5
  2. 5
  3. 4
Extended-level mixed set Extended only
  1. Two similar shapes have a length scale factor of 1.5. The smaller area is 40 cm². Find the larger area.
  2. Two similar solids have volumes 64 cm³ and 216 cm³. Find the length scale factor.
  3. How many planes of symmetry does a non-cube cuboid have?
Show answers
  1. 1.5²=2.25. 40×2.25 = 90 cm²
  2. 216/64=3.375. ³√3.375 = 1.5
  3. 3

Practice at Home

A large set again — take it steadily.

Part A — Length scale factor
  1. Two similar triangles have bases 6 cm and 21 cm. The smaller triangle’s height is 4 cm. Find the larger triangle’s height.
  2. Two similar shapes have scale factor 4. A side of 3.5 cm on the smaller shape corresponds to what length on the larger shape?
Part B — Area and volume scale factor Extended
  1. Two similar shapes have length scale factor 5. The smaller area is 6 cm². Find the larger area.
  2. Two similar solids have length scale factor 3. The smaller volume is 10 cm³. Find the larger volume.
  3. Two similar shapes have areas 16 cm² and 81 cm². Find the length scale factor.
  4. Two similar solids have volumes 125 cm³ and 8 cm³. Find the length scale factor.
Part C — Proving similarity Extended
  1. In triangle PQR, S lies on PQ and T lies on PR, with ST parallel to QR. Explain why triangle PST is similar to triangle PQR.
  2. Using the triangle in question 1, if PS = 4 cm, PQ = 10 cm, and ST = 6 cm, find QR.
Part D — Symmetry
  1. How many lines of symmetry does a regular decagon (10 sides) have?
  2. What is the order of rotational symmetry of a regular hexagon?
  3. How many planes of symmetry does a cube have? Ext.
  4. Does a scalene triangle have any rotational symmetry (order greater than 1)?
Part E — Applied / challenge
  1. A model car is built at a scale of 1:20. If the real car’s volume is 6 m³, find the model’s volume in cm³ (convert units carefully).
  2. Two similar cylindrical tanks have heights 2 m and 5 m. The smaller tank holds 800 litres. Find the capacity of the larger tank.
  3. Explain why a circle has infinitely many lines of symmetry and infinite rotational symmetry, using the definitions from this note.

Key Vocabulary Recap

Similar — same shape and angles, sides in the same ratio
Scale factor — the multiplier connecting corresponding lengths
Line of symmetry — a fold line where both halves match exactly
Order of rotational symmetry — how many times a shape matches itself in one full turn
Plane of symmetry — a flat “mirror” slicing a solid into identical halves Ext.

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) sections C4.4 and C4.5 (Core subject content) and sections E4.4 and E4.5 (Extended subject content) — Similarity, and Symmetry. Calculating lengths of similar shapes, line symmetry, and rotational symmetry are shared by both tiers. Area and volume scale factors, proving similarity using geometric reasons, and symmetry properties of 3D solids are Extended-only, marked above.

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