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GCSE Maths · Geometry and measures
Symmetry and properties of shapes GCSE Questions and Worked Answers
A line of symmetry reflects a shape onto itself. Rotational symmetry counts the matches during a full turn. Use stated side and angle properties, not how a sketch happens to look.
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Start with the meaning
What you need to know about symmetry and properties of shapes
Cut a rectangle from paper and fold it so that the left half exactly covers the right. The crease is a line of symmetry: every point has a matching point on the other side at the same perpendicular distance. Folding top onto bottom gives another line. A diagonal fold does not match a non-square rectangle's edges. This is a test of the whole shape, not just whether a line divides its area in half.
See the idea first
Matching halves are stronger than equal areas
The square has four matching folds; the non-square rectangle has two. Turning the rectangle through 180° also makes it match itself. It matches twice in a full 360° turn, including the completed turn: its rotational order is 2.
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Find reflection symmetry
What the problem asks: Draw the symmetry lines of a square.
How to solve it: Draw the two diagonals and the two lines through opposite side midpoints. Each reflects the whole square onto itself.
Find rotational order
What the problem asks: A regular pentagon matches itself every 72°. Find its order.
How to solve it: 360 ÷ 72 = 5 matches in a full turn. Count the match at 360°, not an extra starting match at 0°.
Identify a quadrilateral
What the problem asks: A quadrilateral has four equal sides but no right angles. What is it?
How to solve it: It is a non-square rhombus. Equal sides alone do not guarantee a square; four right angles would be needed too.
A reliable routine
Test a claimed symmetry or shape property
Use matching points for symmetry and definitions for classification. There is no single calculation that identifies every shape.
- For a proposed fold, check equal perpendicular distances on opposite sides of the line.
- For rotations, find the smallest positive matching turn; divide 360° by it.
- For quadrilaterals, check equal sides, parallel sides and right angles separately.
- Use inclusive definitions: a square is also a rectangle, a rhombus and a parallelogram.
Check: An ordinary shape with no non-trivial rotational symmetry still has order 1 because it matches after 360°. A circle has infinitely many symmetry lines and matches after any rotation. Here a trapezium has exactly one pair of parallel sides; state the convention if a question uses a different definition.
Fully worked
Symmetry and properties of shapes GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
Rectangle symmetry
Question
State the number of symmetry lines and rotational order of a non-square rectangle.
It has 2 lines through opposite side midpoints and rotational order 2. A quarter-turn swaps unequal lengths, so does not match.
Example 2
Regular polygon
Question
Find the symmetry-line count, rotational order and smallest matching turn of a regular hexagon.
Six equivalent sides and corners give 6 lines and order 6.
Example 3
Rhombus
Question
A rhombus is not a square. State its reflection and rotational symmetry.
Its 2 diagonals are symmetry lines. Opposite vertices swap in a half-turn, giving order 2. Its diagonals are unequal; equal diagonals would make this rhombus a square.
Example 4
Parallelogram
Question
A parallelogram has neither equal adjacent sides nor right angles. State its symmetries.
It has no reflection symmetry and rotational order 2. A half-turn swaps both pairs of opposite sides. Equal-area diagonal halves are not mirror halves.
Example 5
Use a property
Question
A parallelogram has one angle of 68°. Find its other angles.
Opposite angles are equal; adjacent angles add to 180° because the sides are parallel.
Other angles: 112°, 68°, 112°.
Example 6
Triangle classification
Question
A triangle has angles 50°, 50° and 80°. Explain its side and symmetry properties.
The two equal angles face two equal sides, so it is isosceles. It has one symmetry line through the 80° vertex and opposite side midpoint, and rotational order 1.
10 original questions · total 20 marks
Symmetry and properties of shapes GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 25 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
Square
State a square's number of symmetry lines and rotational order.
Show worked answer
4 lines; order 4. Its smallest matching turn is 90°.
Equilateral triangle
Find the smallest positive matching rotation of an equilateral triangle.
Show worked answer
Order 3, so
Regular octagon
How many symmetry lines has a regular octagon?
Show worked answer
8: four through opposite vertices and four through opposite side midpoints.
Order from angle
A shape's smallest positive matching turn is 45°. Find its rotational order.
Show worked answer
Scalene triangle
State the symmetries of a scalene triangle.
Show worked answer
All sides differ, so there is no symmetry line. Rotational order is 1.
Kite
A convex kite has two distinct pairs of adjacent equal sides, but not all four sides equal. State its symmetries.
Show worked answer
It has one line of symmetry, through the vertices where each equal pair meets, and rotational order 1.
Square classification
Is every square a rectangle? Explain.
Show worked answer
Yes. A rectangle is a quadrilateral with four right angles, and a square has those. The extra equal-side property does not remove it from the rectangle family.
Equal sides are not enough
A pupil says every rhombus is a square. Give a counterexample using angles.
Show worked answer
A rhombus with angles 60°, 120°, 60°, 120° has equal sides but no right angles, so is not a square.
Isosceles trapezium
An isosceles trapezium has exactly one pair of parallel sides and equal non-parallel sides. State its reflection symmetry.
Show worked answer
It has one symmetry line, perpendicular to both parallel sides and through their midpoints.
Circle
How many symmetry lines does a circle have?
Show worked answer
Infinitely many: every line through its centre is a symmetry line.
Examiner-style feedback
Common symmetry and properties of shapes mistakes
A fold must match every part of the shape.
Use given measurements or equal-side marks.
A full turn always matches a bounded shape, so the minimum order is 1.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
- Test the whole shape.
- Count matches in one full turn.
- Use definitions, not appearance.
- Regular means equal sides and equal angles.
Quick answers
Symmetry and properties of shapes FAQ
Is a square a rhombus?
Yes. It has four equal sides, and also satisfies the extra right-angle condition.
Does a parallelogram always have reflection symmetry?
No. Special parallelograms such as rectangles and rhombi do; a general one need not.
Content standards
Curriculum and rights review
G1/G4 shape definitions, triangle and quadrilateral properties and symmetry across tiers. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
Official specification references