Geometry Toolkit Notes

Cambridge IGCSE Maths: Revision notes

Key facts

  • Angles are named by size: acute (below 90°), right (90°), obtuse (90° to 180°) and reflex (180° to 360°).
  • Similar shapes have equal angles and proportional sides; congruent shapes are identical in shape and size.
  • Triangles: equilateral, isosceles, scalene and right-angled. Know the six special quadrilaterals and their properties.
  • Circle words: radius, diameter, chord, tangent, arc, sector and segment.
  • Use a ruler for every straight line and a protractor for every angle, and label your diagram.

Angle and line words

Geometry questions assume you know the vocabulary of points, lines and angle sizes.

A vertex is a corner where lines or edges meet. Parallel lines never meet; perpendicular lines meet at 90°. A perpendicular bisector cuts a line in half at a right angle. A bearing is a clockwise angle from north. An interior angle is inside a shape; an exterior angle lies between a side and the extension of the next side.

40°130°OABCD
From the ray OA: an acute angle (40°), a right angle (90°) and an obtuse angle (130°). A reflex angle is more than 180°.

Acute

  • Less than 90°

Right angle

  • Exactly 90°

Obtuse

  • Between 90° and 180°

Reflex

  • Between 180° and 360°

Similar

  • Same shape
  • Equal angles
  • Sides in proportion

Congruent

  • Same shape and size
  • Equal angles
  • Equal sides

An angle measures 200°. What type of angle is it?

Triangles and quadrilaterals

Triangles are named by their sides and angles; each special quadrilateral has its own properties.

An equilateral triangle has three equal sides and three 60° angles. An isosceles triangle has two equal sides and two equal base angles. A scalene triangle has no equal sides. A right-angled triangle contains a 90° angle.

6 cm5 cm5 cmABC
An isosceles triangle: two equal sides (5 cm) and equal base angles

Square

  • 4 equal sides
  • 4 right angles

Rectangle

  • 2 pairs of equal sides
  • 4 right angles

Rhombus

  • 4 equal sides
  • Opposite angles equal

Parallelogram

  • 2 pairs of parallel, equal sides

Kite

  • 2 pairs of adjacent equal sides

Trapezium

  • Exactly one pair of parallel sides

Which quadrilateral has exactly one pair of parallel sides?

Polygons, nets and solids

Polygons are named by their number of sides, and nets are flat patterns that fold into solids.

A regular polygon has all sides and angles equal. A net is a 2D shape that folds into a 3D solid, and is used to find surface area. Know these solids: cube, cuboid, prism, cylinder, pyramid, cone and sphere. A frustum is what remains when the top of a cone or pyramid is cut off parallel to its base. A hemisphere is not needed at Core.

3 cm120°ABCDEF
A regular hexagon: 6 equal sides and 6 equal interior angles of 120°

Pentagon

  • 5 sides

Hexagon

  • 6 sides

Octagon

  • 8 sides

Decagon

  • 10 sides

How many sides does an octagon have?

Parts of a circle

Each part of a circle has its own name, and exam questions use them without explanation.

The radius runs from the centre to the circumference. The diameter is a chord through the centre, twice the radius. A chord joins two points on the circumference. A tangent touches the circle at exactly one point. An arc is part of the circumference. A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc.

diameterradiuschordtangentOABCDE
Parts of a circle: radius, diameter, chord and tangent

Which part of a circle touches it at exactly one point?

Measuring and drawing

Accurate diagrams need a ruler, a protractor and clear labels.

Draw every straight edge with a ruler, never freehand. Measure and draw angles with a protractor. Label every given length and angle. Give angles to 1 decimal place and lengths to the nearest millimetre. Sketching a labelled diagram from a worded description is worth doing even when no drawing is asked for.

  1. 1

    Sketch

    Rough labelled diagram first

  2. 2

    Ruler

    Draw each straight edge

  3. 3

    Protractor

    Measure or draw each angle

  4. 4

    Label

    Add all lengths and angles

Drawing a diagram from a description

Which tool should you use to draw an angle accurately?

Try an exam question

(a) Write down the name of a triangle with exactly two equal sides. (b) Write down the name of a line that touches a circle at one point. (c) Explain the difference between similar shapes and congruent shapes.

[4 marks]

That's the notes covered.

Carry on to the next subtopic.