Properties of 2D Shapes Notes

Edexcel GCSE Maths: Revision notes

Key facts

  • Parallel lines never meet; perpendicular lines meet at 90°.
  • A regular polygon has all sides and all angles equal.
  • A square is a special rectangle and a special rhombus.
  • Rotational symmetry of order nn looks the same nn times in a full turn.
  • Interior angles add up to (n−2)×180∘(n-2)\times180^\circ; each exterior angle of a regular polygon is 360∘n\dfrac{360^\circ}{n}.

Lines, angles and polygons

Use exact vocabulary: a polygon is a closed 2D shape with straight sides, and regular means all sides and angles are equal.

Parallel lines never meet; perpendicular lines meet at 90°.

A polygon has straight sides; its corners are vertices. A regular polygon has equal sides and equal angles.

A diagonal joins two vertices that are not next to each other.

Triangles: equilateral (3 equal sides), isosceles (2 equal sides), scalene (none), right-angled (one 90°).

ABCDE
A regular pentagon ABCDE with its five diagonals.

Triangle

  • 3 sides

Quadrilateral

  • 4 sides

Pentagon

  • 5 sides

Hexagon

  • 6 sides

What is a diagonal of a polygon?

Special quadrilaterals

Each special quadrilateral has its own set of side, angle and parallel-line properties.

A trapezium has exactly one pair of parallel sides.

A kite has two pairs of equal adjacent sides and one pair of equal opposite angles.

Use the table for the four parallelogram types. In a rhombus the diagonals cross at right angles; in a rectangle they are equal.

ABCDEFGHIJKM
A parallelogram, a trapezium and a kite (left to right).

Square

All sides equal:
Yes
All angles 90°:
Yes
Opposite sides parallel:
Yes
Opposite angles equal:
Yes

Rectangle

All sides equal:
No
All angles 90°:
Yes
Opposite sides parallel:
Yes
Opposite angles equal:
Yes

Rhombus

All sides equal:
Yes
All angles 90°:
No
Opposite sides parallel:
Yes
Opposite angles equal:
Yes

Parallelogram

All sides equal:
No
All angles 90°:
No
Opposite sides parallel:
Yes
Opposite angles equal:
Yes

Which quadrilateral has exactly one pair of parallel sides?

Symmetry

A shape has line symmetry if it folds onto itself, and rotational symmetry if it fits itself during a turn.

A line of symmetry splits a shape into two matching halves.

Rotational symmetry of order nn: the shape looks identical nn times in one full turn, each turn being 360∘n\dfrac{360^\circ}{n} about the centre.

A square has 4 lines of symmetry and order 4. An equilateral triangle has 3 lines and order 3.

ABCD
A square has 4 lines of symmetry (the two diagonals and two lines through midpoints).

Square

Lines of symmetry:
4
Order of rotation:
4

Rectangle

Lines of symmetry:
2
Order of rotation:
2

Equilateral triangle

Lines of symmetry:
3
Order of rotation:
3

Regular hexagon

Lines of symmetry:
6
Order of rotation:
6

What is the order of rotational symmetry of an equilateral triangle?

Angles in polygons

The interior angles of an nn-sided polygon add up to (n−2)×180∘(n-2)\times180^\circ.

Interior and exterior angles at a vertex add up to 180∘180^\circ.

For a regular polygon with nn sides:

  • each exterior angle =360∘n=\dfrac{360^\circ}{n}
  • each interior angle =180∘−exterior angle=180^\circ-\text{exterior angle}

Exterior angles of any polygon add up to 360∘360^\circ.

120°60°ABCDEF
Regular hexagon: interior angle 120° at A, exterior angle 60° at B.
  • Sum of interior angles(n−2)×180∘(n-2)\times180^\circ
  • Exterior angle (regular)360∘n\dfrac{360^\circ}{n}

Worked example

Find each interior and exterior angle of a regular hexagon.

What is each interior angle of a regular octagon?

Try an exam question

ABCDEFGH is a regular octagon. Work out the size of one interior angle.

[3 marks]

That's the notes covered.

Carry on to the next subtopic.