Properties of shapes | AQA GCSE Maths Notes

Visual model

Shape properties help you choose the right rule

rectangletrianglepentagonname the shape, then choose the rule

Gold-standard guide

20 mins

What you will learn

Recognise and reason with 2D and 3D shapes.

Use a clear step-by-step method for properties of shapes.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier geometry and measures skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Learn the hierarchy: square → rectangle → parallelogram → quadrilateral

Step 1

State the four properties of a parallelogram

Step 2

Explain how a rectangle differs

A rectangle has all the parallelogram properties PLUS all four angles are 90° and diagonals are equal in length

Step 3

Note the implication

Every rectangle is a parallelogram, but not every parallelogram is a rectangle

Watch out

Students list properties of shapes without stating them precisely

Interior angles

$sum of interior angles = (n - 2) \times 180 degrees.$

Exterior angle

$regular polygon exterior angle = 36\frac{0}{n}.$

Worked example

List four properties of a parallelogram and explain how a rectangle differs from it.

State the four properties of a parallelogram: 1. Opposite sides are parallel and equal. 2. Opposite angles are equal.3. Diagonals bisect each other. 4. Two pairs of parallel sides.

Explain how a rectangle differs: A rectangle has all the parallelogram properties PLUS all four angles are 90° and diagonals are equal in length.

Note the implication: Every rectangle is a parallelogram, but not every parallelogram is a rectangle.

Final answer

Parallelogram: opp sides equal and parallel, opp angles equal, diagonals bisect. Rectangle adds: all 90°, equal diagonals.

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

List four properties of a parallelogram and explain how a rectangle differs from it.

3 marks4 minsproperties-of-shapes-worked

Show solution

Worked solution

1.State the four properties of a parallelogram: 1. Opposite sides are parallel and equal. 2. Opposite angles are equal.3. Diagonals bisect each other. 4. Two pairs of parallel sides.
2.Explain how a rectangle differs: A rectangle has all the parallelogram properties PLUS all four angles are 90° and diagonals are equal in length.
3.Note the implication: Every rectangle is a parallelogram, but not every parallelogram is a rectangle.

Final answer

Parallelogram: opp sides equal and parallel, opp angles equal, diagonals bisect. Rectangle adds: all 90°, equal diagonals.

Mark points

M1: state the four properties of a parallelogram
M1: explain how a rectangle differs
M1: note the implication
A1: Parallelogram: opp sides equal and parallel, opp angles equal, diagonals bisect.Rectangle adds: all 90°, equal diagonals.

Watch out

Students list properties of shapes without stating them precisely.'Opposite sides are equal' is not enough — you must say 'and parallel' for a parallelogram.Use exact geometric language in every description.

Diagnosticrecall

Name a quadrilateral with exactly one pair of parallel sides.

1 mark2 minsproperties-of-shapes-q1

Show solution

Worked solution

1.Spot the skill: Learn the hierarchy: square → rectangle → parallelogram → quadrilateral.
2.Use the state the four properties of a parallelogram stage first, then explain how a rectangle differs.
3.Keep the final answer visible: Trapezium.

Final answer

Trapezium

Mark points

M1: use the correct learn the hierarchy: square → rectangle → parallelogram → quadrilateral.higher up means more properties. shapes sharing a parent also share its properties.for 3d shapes, name faces, edges and vertices.
A1: Trapezium

Watch out

Easyprocedure

How many lines of symmetry does a regular pentagon have?

2 marks3 minsproperties-of-shapes-q2

Show solution

Worked solution

1.Spot the skill: Learn the hierarchy: square → rectangle → parallelogram → quadrilateral.
2.Use the explain how a rectangle differs stage first, then note the implication.
3.Keep the final answer visible: 5.

Final answer

Mark points

M1: use the correct learn the hierarchy: square → rectangle → parallelogram → quadrilateral.higher up means more properties. shapes sharing a parent also share its properties.for 3d shapes, name faces, edges and vertices.
A1: 5

Watch out

Mediumreasoning

Name all 2D shapes with equal diagonals that bisect each other.

3 marks4 minsproperties-of-shapes-q3

Show solution

Worked solution

1.Spot the skill: Learn the hierarchy: square → rectangle → parallelogram → quadrilateral.
2.Use the note the implication stage first, then state the four properties of a parallelogram.
3.Keep the final answer visible: Rectangle, square.

Final answer

Rectangle, square

Mark points

M1: use the correct learn the hierarchy: square → rectangle → parallelogram → quadrilateral.higher up means more properties. shapes sharing a parent also share its properties.for 3d shapes, name faces, edges and vertices.
A1: Rectangle, square

Watch out

Hardproblem solving

A cuboid has dimensions 3 × 4 × 5 cm. State faces, edges and vertices.

3 marks5 minsproperties-of-shapes-q4

Show solution

Worked solution

1.Spot the skill: Learn the hierarchy: square → rectangle → parallelogram → quadrilateral.
2.Use the state the four properties of a parallelogram stage first, then explain how a rectangle differs.
3.Keep the final answer visible: 6 faces, 12 edges, 8 vertices.

Final answer

6 faces, 12 edges, 8 vertices

Mark points

M1: use the correct learn the hierarchy: square → rectangle → parallelogram → quadrilateral.higher up means more properties. shapes sharing a parent also share its properties.for 3d shapes, name faces, edges and vertices.
A1: 6 faces, 12 edges, 8 vertices

Watch out

Exam-stylemulti-step

What is the order of rotational symmetry of a regular hexagon?

4 marks6 minsproperties-of-shapes-q5

Show solution

Worked solution

1.Spot the skill: Learn the hierarchy: square → rectangle → parallelogram → quadrilateral.
2.Use the explain how a rectangle differs stage first, then note the implication.
3.Keep the final answer visible: 6.

Final answer

Mark points

M1: use the correct learn the hierarchy: square → rectangle → parallelogram → quadrilateral.higher up means more properties. shapes sharing a parent also share its properties.for 3d shapes, name faces, edges and vertices.
A1: 6

Watch out

Grade 9 stretchproblem solving

Explain why every square is both a rectangle and a rhombus.

4 marks7 minsshape-g9

Show solution

Worked solution

1.State the defining property of a rectangle.
2.State the defining property of a rhombus.
3.Show that a square satisfies both.

Final answer

A square has four right angles and four equal sides, so it meets both definitions.

Mark points

C1: rectangle property
C1: rhombus property
C1: conclusion

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Properties of shapes - 2 marksName a quadrilateral with exactly one pair of parallel sides.Mark answer

Answer

Trapezium

2Angles, lines and polygons - 2 marksThe exterior angle of a regular polygon is 24°. How many sides?Mark answer

Answer

3Perimeter, area and volume - 2 marksA prism has cross-section area 24 cm² and length 7 cm. Find its volume.Mark answer

Answer

168 cm³

4Circles - 3 marksFind the diameter of a circle with area

100pi c

m².Mark answer

Answer

20 cm

Mastery check

I can explain the method for properties of shapes.
I can show clear working without skipping key steps.
I can avoid this mistake: Students list properties of shapes without stating them precisely.'Opposite sides are equal' is not enough — you must say 'and parallel' for a parallelogram.Use exact geometric language in every description.

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