Geometrical Constructions

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Measure and draw lines and angles
  2. Construct a triangle, given the lengths of all sides, using a ruler and pair of compasses only
  3. Draw, use and interpret nets
  4. Understand that construction arcs must be shown and that a ruler should be used for all straight edges

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1. Measuring and Drawing Lines and Angles

Measuring and Drawing Lines

A line is a straight path between two points. To measure and draw lines accurately, you need a ruler.

How to measure a line:

  • Place the ruler so that the zero mark lines up with one end of the line
  • Read the measurement where the other end of the line meets the ruler
  • Always use the correct units (usually centimetres or millimetres)

How to draw a line of a specific length:

  • Mark your starting point on the paper
  • Place the ruler with the zero mark at your starting point
  • Make a small mark at the required length
  • Use the ruler to draw a straight line connecting the two points

Important tips:

  • Keep your ruler still while drawing
  • Use a sharp pencil for accurate measurements
  • Always use a ruler for straight edges – never draw freehand lines in constructions

Measuring and Drawing Angles

An angle is the amount of turn between two lines that meet at a point (called the vertex).

How to measure an angle:

  • Place the protractor's centre point (the small circle or cross) exactly on the vertex of the angle
  • Line up the base line of the protractor with one arm of the angle
  • Read the measurement where the other arm crosses the protractor scale
  • Make sure you read the correct scale (protractors usually have two scales going in opposite directions)

How to draw an angle of a specific size:

  • Draw a base line using your ruler
  • Mark the vertex (the corner point of the angle)
  • Place the protractor's centre on the vertex and line up the base line with 0°
  • Make a small mark at the required angle measurement
  • Remove the protractor and use your ruler to draw a straight line from the vertex through your mark

2. Constructing a Triangle Using a Ruler and Pair of Compasses

When you construct a triangle, you must use only a ruler and a pair of compasses. This means you cannot use a protractor to measure angles.

What are compasses? Compasses are a drawing tool with two legs – one has a pointed end and the other holds a pencil. They are used to draw circles and arcs (parts of circles).

How to Construct a Triangle When You Know All Three Side Lengths

Let's say you need to construct a triangle with sides of 6 cm, 7 cm, and 8 cm.

Step-by-step method:

  1. Draw the first side (the base)

    • Use your ruler to draw a straight line 8 cm long (this will be one side of the triangle)
    • Label the ends A and B

  2. Set your compasses for the second side

    • Open your compasses to 6 cm (you can measure this against your ruler)
    • Place the compass point on point A
    • Draw an arc (a curved line) above the base line
    • Important: Do NOT rub out this arc – construction arcs must be shown in your final answer

  3. Set your compasses for the third side

    • Open your compasses to 7 cm
    • Place the compass point on point B
    • Draw another arc that crosses the first arc
    • The point where the two arcs cross is the third vertex of your triangle

  4. Complete the triangle

    • Label the point where the arcs cross as point C
    • Use your ruler to draw a straight line from A to C
    • Use your ruler to draw a straight line from B to C
    • You now have a triangle with sides 8 cm (AB), 6 cm (AC), and 7 cm (BC)

Why must construction arcs be shown? The arcs show the method you used to construct the triangle. In exams, you must leave them visible to prove you used the correct construction technique and didn't just guess the position or use a protractor.

Constructing Other Shapes Using Triangles

You can use the same triangle construction method to build other shapes. For example, to construct a rhombus (a four-sided shape where all sides are equal length):

  1. Construct one triangle using the method above
  2. Construct a second identical triangle on the other side of the shared base
  3. The two triangles together form a rhombus

Remember: A rhombus has four equal sides but the angles are not necessarily 90°.

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