Geometry and measures

Properties of 2D shapes and symmetry

10 past-paper questions, easiest first. Try each one, then reveal the answer.

Foundation & HigherAll 10 from Foundation papers

Revision guide by The Full Marks

Properties of 2D shapes and symmetry: what you need to know

This topic is about naming shapes and knowing their properties: sides, angles, diagonals and symmetry. These marks are some of the easiest on a paper, and the same facts unlock harder angle questions later on.

Key facts

Parallel lines never meet (marked with arrows); perpendicular lines meet at $90^\circ$ .
Square: 4 equal sides, 4 right angles, 4 lines of symmetry, rotational symmetry order 4. Rectangle: 2 lines, order 2.
Parallelogram: opposite sides equal and parallel, opposite angles equal, no lines of symmetry, order 2. Rhombus: a parallelogram with 4 equal sides, 2 lines, order 2.
Trapezium: exactly one pair of parallel sides. Kite: two pairs of equal adjacent sides, one line of symmetry, one pair of equal opposite angles.
Isosceles triangle: 2 equal sides and 2 equal base angles. Equilateral: 3 equal sides, every angle $60^\circ$ . Scalene: no equal sides. Right-angled: one angle of $90^\circ$ .
Diagonals: they cross at right angles in a square, rhombus and kite, and are equal in length in a square and rectangle. A regular polygon has all sides equal and all angles equal.
Angle $ABC$ means the angle at $B$ , between lines $BA$ and $BC$ .

Identifying a shape from its properties

Count the sides: 5 pentagon, 6 hexagon, 7 heptagon, 8 octagon, 9 nonagon, 10 decagon.
For a quadrilateral, check parallel sides first, then equal sides, then right angles.
For symmetry, count mirror lines (lines of symmetry) and how many times it fits onto itself in one full turn (order of rotational symmetry).

Worked examples

Example 1 Foundation · grade 3

$ABCD$ is a kite with $AB = AD$ and $CB = CD$ . Angle $BAD = 40^\circ$ and angle $BCD = 100^\circ$ . Find angle $ABC$ .

Show the solution

Angles in a quadrilateral add up to $360^\circ$ .
$360 - 40 - 100 = 220$
A kite has one pair of equal opposite angles, so angle $ABC$ = angle $ADC$ .
$220 \div 2 = 110$
Angle $ABC = 110^\circ$ .

Example 2 Foundation · grade 4

$ABCD$ is a rhombus. Angle $ABC = 64^\circ$ . Find angle $BAC$ .

Show the solution

In a rhombus all sides are equal, so $AB = BC$ and triangle $ABC$ is isosceles.
Base angles of an isosceles triangle are equal, so angle $BAC$ = angle $BCA$ .
Angles in a triangle add up to $180^\circ$ .
$180 - 64 = 116$
$116 \div 2 = 58$
Angle $BAC = 58^\circ$ .

Common mistakes

Calling a rhombus or parallelogram a rectangle, or a trapezium a parallelogram. Check for right angles and count the pairs of parallel sides before you use the name.
Saying a rectangle or parallelogram has diagonal lines of symmetry. Fold it in your head: the halves do not match.
Mixing up parallel (same direction, never meet) and perpendicular (meet at a right angle).
Reading the wrong scale on a protractor. Decide first whether the angle is acute or obtuse, then check your reading matches.

Exam tips

Use the proper mathematical name, for example rhombus, not 'diamond'.
When asked to draw a shape with given symmetry, check both conditions: the number of mirror lines and the order of rotation.

Questions people ask

What is the difference between a rhombus and a parallelogram?

A rhombus is a special parallelogram with all four sides the same length. Both have opposite sides parallel and opposite angles equal.

How many lines of symmetry does a regular polygon have?

The same number as its sides. A regular hexagon has 6 lines of symmetry and rotational symmetry of order 6.

Exam questions: Properties of 2D shapes and symmetry (10, easiest first)

Properties of 2D shapes and symmetry1 markGrade 1CalculatorFoundation

November 2024 · Paper 2 Foundation · Q4

Show answer

Answer	hexagon	1 mark

Worked solution Checked against the mark scheme

Count the sides: there are 6.

A polygon with 6 sides is a hexagon.

Watch out: Confusing hexagon (6 sides) with pentagon or octagon.

Properties of 2D shapes and symmetry3 marksGrade 1CalculatorFoundation

June 2024 · Paper 2 Foundation · Q7

Measure a line and an angle, and draw a six-sided shape.

Show answer

(a)	$8.7$ cm (accept 8.5 to 8.9)	1 mark
(b)	$67^\circ$ (accept 65 to 69)	1 mark
(c)	Any 6-sided polygon	1 mark

Worked solution Checked against the mark scheme

(a) Line up the ruler's zero with one end of the line and read the other end: about $8.7$ cm.

(b) Put the protractor's centre on the vertex and a zero line along one arm. The angle is acute, so read the smaller scale: about $67^\circ$ .

Watch out: Reading the wrong protractor scale and getting 113°.

Properties of 2D shapes and symmetry2 marksGrade 1CalculatorFoundation

November 2024 · Paper 3 Foundation · Q7

Name a four-sided shape on a grid and draw a triangle containing a right angle.

Show answer

(a)	Trapezium	1 mark
(b)	Any right-angled triangle drawn on the grid	1 mark

Worked solution Checked against the mark scheme

(a) The shape has exactly one pair of parallel sides (top and bottom), so it is a trapezium.

(b) Draw along one horizontal grid line and one vertical grid line from the same point, then join their ends. The grid lines meet at $90°$ .

Watch out: Calling the shape a parallelogram.

Properties of 2D shapes and symmetry2 marksGrade 1CalculatorFoundation

June 2023 · Paper 3 Foundation · Q8

Name a quadrilateral from its properties and name a drawn 3D solid.

Show answer

(a)	square	1 mark
(b)	cuboid	1 mark

Worked solution Checked against the mark scheme

(a) Four right angles and four equal sides make a square.

(b) The solid has six rectangular faces, so it is a cuboid.

Watch out: Calling the shape in (a) a rectangle, or the solid a cube.

Properties of 2D shapes and symmetry2 marksGrade 1No calculatorFoundation

November 2021 · Paper 1 Foundation · Q6

Name a quadrilateral with one pair of parallel sides and name a 3-D solid with two circular ends.

Show answer

(a)	Trapezium	1 mark
(b)	Cylinder	1 mark

Worked solution Checked against the mark scheme

(a) The arrows show exactly one pair of parallel sides, so it is a trapezium.

(b) A solid with two equal circular faces joined by a curved surface is a cylinder.

Watch out: Calling the trapezium a parallelogram.

Properties of 2D shapes and symmetry3 marksGrade 2CalculatorFoundation

June 2023 · Paper 2 Foundation · Q7

Measure a side and an angle of a drawn triangle, then name a triangle with two equal sides.

Show answer

(a)	9.3 cm (9.1 to 9.5 accepted)	1 mark
(b)	$106^\circ$ (104 to 108 accepted)	1 mark
(c)	Isosceles	1 mark

Worked solution Checked against the mark scheme

(a) Put the ruler's 0 on A and read off at C: about $9.3$ cm.

(b) Place the protractor's centre on B, line up one side with 0 and read the scale that starts at 0: about $106^\circ$ . It is obtuse, so the answer must be over $90^\circ$ .

Watch out: Reading the wrong protractor scale and giving about $74^\circ$ .

Properties of 2D shapes and symmetry3 marksGrade 2CalculatorFoundation

June 2022 · Paper 2 Foundation · Q7

Name a six-sided polygon drawn on a grid and identify parallel and perpendicular sides.

Show answer

(a)	Hexagon	1 mark
(b)	$AF$	1 mark
(c)	$AB$ or $EF$	1 mark

Worked solution Checked against the mark scheme

(a) The shape has 6 sides, so it is a hexagon.

(b) $CD$ is horizontal. The other horizontal side is $AF$ .

Watch out: Mixing up parallel and perpendicular.

Properties of 2D shapes and symmetry2 marksGrade 3CalculatorFoundation

June 2022 · Paper 3 Foundation · Q6

Draw on a grid a four-sided shape that has rotational symmetry of order 2 but no lines of symmetry.

Show answer

Answer	A parallelogram that is not a rectangle or rhombus	2 marks

Worked solution Checked against the mark scheme

A rectangle and a rhombus have order 2 but also have lines of symmetry, so avoid them.

A general parallelogram (slanted, with unequal adjacent sides) has order 2 and no lines of symmetry. Draw one on the grid.

Watch out: Drawing a rectangle, which has two lines of symmetry.

Properties of 2D shapes and symmetry3 marksGrade 3CalculatorFoundation

June 2019 · Paper 2 Foundation · Q13

Find the width of a rectangle that contains a staircase of identical squares, given its length.

Show answer

Answer	10 cm	3 marks

Worked solution Checked against the mark scheme

Across the length there are 6 squares side by side, so each side is $12 \div 6 = 2$ cm.

Down the width there are 5 squares, so the width is $5 \times 2 = 10$ cm.

Watch out: Dividing 12 by 9 because there are nine squares.

Properties of 2D shapes and symmetry2 marksGrade 4CalculatorFoundation

June 2019 · Paper 3 Foundation · Q20

Find an angle made by a square and an equilateral triangle sharing a side.

Show answer

Answer	105	2 marks

Worked solution Checked against the mark scheme

EB is a diagonal of the square, so it cuts the right angle at B in half: angle $EBD = 45^\circ$ .

The triangle is equilateral, so angle $DBC = 60^\circ$ .

Angle $EBC = 45 + 60 = 105^\circ$

Watch out: Using 90° for angle EBD.

Also tested in these questions

Their main topic is different, but part of each one tests this topic.

November 2024 · Paper 1 Foundation · Q7: Measure a side and an angle of an accurately drawn triangle and name the angle type.
June 2024 · Paper 3 Foundation · Q20: Use isosceles triangle angle facts to find two angles and write them as a ratio 1 : n.
June 2023 · Paper 1 Foundation · Q8: Find a reflex angle at a point and explain why a given angle is not obtuse.
June 2022 · Paper 3 Foundation · Q16: Draw an isosceles triangle with a given area on a centimetre grid.
November 2019 · Paper 2 Foundation · Q17: Use isosceles triangles on a straight line to find an angle, giving reasons.
November 2018 · Paper 1 Foundation · Q14: Use angle facts to find the angles of a quadrilateral and prove it is a kite.
November 2018 · Paper 3 Foundation · Q9: Name one shape on a grid and pick out the two congruent shapes.
June 2018 · Paper 1 Foundation · Q24: Use the coordinates of two corners of a stepped pattern of four equal squares to find another marked point.
June 2018 · Paper 2 Foundation · Q15: Explain the errors in two angle arguments: an isosceles triangle and angles on parallel lines.
Specimen · Paper 2 Foundation · Q17: Use angle facts in a right-angled triangle and an isosceles triangle to find an angle, with reasons.
Specimen · Paper 3 Foundation · Q17: Find the three possible base or apex angles of an isosceles triangle with one 70° angle, and explain why a 120° angle gives only one option.