IGCSE Maths · Geometry · Core & Extended

Constructions and Scale Drawings: Building Accurate Shapes

This topic is identical for Core and Extended — every skill here is examined the same way at both tiers. It’s also the most “hands-on” part of Geometry: measuring accurately, constructing with a compass, reading scale drawings, and finding bearings all reward careful, precise work over quick estimation.

1. Measuring and Drawing Lines and Angles

A ruler must be used for every straight edge — freehand lines lose marks even if the shape looks roughly right. Measure angles with a protractor, reading from the correct scale (0° on the side you’re measuring from).

Try it — Measuring Accuracy
  1. What tool should always be used for straight edges in a construction?
  2. What tool is used to measure or draw an angle accurately?
  3. Are perpendicular bisector or angle bisector constructions required at either tier?
Show answers
  1. A ruler.
  2. A protractor.
  3. No — the syllabus explicitly excludes these at both Core and Extended.

2. Constructing a Triangle with Ruler and Compasses

Given all three side lengths, you can construct a triangle exactly, without a protractor — draw one side first, then use compass arcs from each end to fix the third vertex.

Method — constructing a triangle with sides 6 cm, 5 cm, 4 cm 6 cm 5 cm 4 cm arcs cross here 1. Draw the base (6 cm) accurately with a ruler.
2. Open the compasses to 5 cm. Place the point on the left end of the base and draw an arc above the line.
3. Open the compasses to 4 cm. Place the point on the right end and draw a second arc, crossing the first.
4. Where the two arcs cross is the third vertex — join it to both ends of the base with a ruler.
The construction arcs must be left visible on your diagram — don’t rub them out, they’re evidence of the method.
Try it — Triangle Construction
  1. Describe the steps to construct a triangle with sides 7 cm, 6 cm, and 5 cm.
  2. Why must construction arcs be left visible rather than erased?
Show answers
  1. Draw a 7 cm base. Set compasses to 6 cm, arc from one end. Set compasses to 5 cm, arc from the other end. Join the base ends to where the arcs cross.
  2. They’re the evidence that a proper compass construction was used, rather than an estimated or measured-and-drawn triangle — marks are usually awarded for showing the method.

3. Nets of 3D Solids

A net is a 2D shape that folds up into a 3D solid. Recognising and drawing nets — and using the measurements on them — connects directly to surface area and volume.

Worked example — a cuboid net 5 × 3 2×3 This net folds into a cuboid measuring 5 cm × 3 cm × 2 cm. Find its surface area and volume using the net’s measurements.
Surface area = 2(5×3) + 2(5×2) + 2(3×2) = 2(15) + 2(10) + 2(6) = 30 + 20 + 12 = 62 cm²
Volume = 5 × 3 × 2 = 30 cm³
Try it — Nets
  1. A cuboid net has faces measuring 4×3, 4×2, and 3×2 (each appearing twice). Find the surface area.
  2. Using the same cuboid, find the volume.
  3. What 3D solid does a net of one rectangle and two circles fold into?
Show answers
  1. 2(12)+2(8)+2(6) = 24+16+12 = 52 cm²
  2. 4×3×2 = 24 cm³
  3. A cylinder.

4. Scale Drawings

A scale drawing represents a real object or place at a fixed ratio — a scale of 1:200 means every 1 unit on the drawing represents 200 of the same unit in real life.

Worked example A floor plan has a scale of 1:200. A bedroom measures 4 cm by 3 cm on the plan. Find its real dimensions and area.
Real length: 4 × 200 = 800 cm = 8 m
Real width: 3 × 200 = 600 cm = 6 m
Real area: 8 × 6 = 48 m²
Try it — Scale Drawings
  1. A scale drawing has a scale of 1:50. A wall measures 8 cm on the drawing. Find its real length in metres.
  2. A scale drawing has a scale of 1:1000. Two buildings are 6.5 cm apart on the drawing. Find the real distance in metres.
Show answers
  1. 8 × 50 = 400 cm = 4 m
  2. 6.5 × 1000 = 6500 cm = 65 m

5. Three-Figure Bearings

A bearing is a direction, always measured clockwise from North, and always written with three digits (so 70° becomes 070°).

Reading a bearing N 070° B A The bearing of B from A is 070° — measured clockwise from North at A, round to the direction of B.
Worked example — finding a back bearing The bearing of B from A is 070°. Find the bearing of A from B.
Since 070° is less than 180°, add 180°: 070 + 180 = 250°
Worked example — the other case The bearing of Y from X is 145°. Find the bearing of X from Y.
Since 145° is less than 180°, add 180°: 145 + 180 = 325°
Try it — Bearings
  1. Write 45° as a three-figure bearing.
  2. The bearing of D from C is 200°. Find the bearing of C from D (careful — this time you subtract, since it’s over 180°).
  3. Point E is due east of point F. State the bearing of E from F.
Show answers
  1. 045°
  2. 200 − 180 = 020°
  3. 090° (due east is always 090°)
Common mistakes to watch for
  • Forgetting to write a bearing with three digits — 70° must be written 070°, not 70°.
  • Measuring a bearing counter-clockwise instead of clockwise from North.
  • For a back bearing, always adding 180° instead of checking whether to add or subtract — if the original bearing is under 180°, add; if it’s 180° or more, subtract.
  • Mixing up the scale direction — multiply a drawing measurement by the scale factor to get the real size, don’t divide.
  • Erasing construction arcs, losing the evidence of a proper compass construction.

6. Quick-Fire Challenge Round

True or False?
  1. Bearings are always written with exactly three digits.
  2. Bearings are measured anticlockwise from North.
  3. To convert a scale drawing measurement to real life, you multiply by the scale factor.
  4. Perpendicular bisector constructions are required at Extended level.
Show answers
  1. True.
  2. False — bearings are measured clockwise from North.
  3. True.
  4. False — not required at either tier.
Spot the Error
  1. “A bearing of 8° should be written as 8°.”
  2. “The bearing of A from B is 300°, so the bearing of B from A is 300 + 180 = 480°.”
  3. “A scale of 1:500 means the real object is 500 times smaller than the drawing.”
Show answers
  1. Bearings always use three digits: it should be written 008°.
  2. Since 300° is over 180°, you subtract 180° instead of adding: 300 − 180 = 120°. (Also, a bearing can never exceed 360°, which is a useful sanity check.)
  3. It’s the other way round — the drawing is 500 times smaller than the real object, not the reverse.

7. Mixed Practice — Bringing It All Together

  1. Describe the steps to construct a triangle with sides 8 cm, 6 cm, and 5 cm.
  2. A scale drawing has scale 1:250. A garden measures 12 cm by 8 cm on the drawing. Find its real area in m².
  3. The bearing of P from Q is 095°. Find the bearing of Q from P.
  4. A cuboid net has faces 6×4, 6×3, and 4×3. Find its surface area.
Show answers
  1. Draw an 8 cm base. Arc of 6 cm from one end, arc of 5 cm from the other. Join the base ends to where the arcs cross.
  2. Real length: 12×250=3000cm=30m. Real width: 8×250=2000cm=20m. Area=30×20=600 m²
  3. 095+180=275°
  4. 2(24)+2(18)+2(12)=48+36+24=108 units²

Practice at Home

A large set again — take it steadily.

Part A — Constructions
  1. Describe the steps to construct a triangle with sides 9 cm, 7 cm, and 6 cm.
  2. Explain why a triangle with sides 3 cm, 4 cm, and 10 cm cannot actually be constructed (hint: think about how far the two shorter arcs could possibly reach).
Part B — Nets
  1. A cube net has all faces 5 cm × 5 cm. Find the cube’s surface area and volume.
  2. A triangular prism net includes two triangular faces (base 6 cm, height 4 cm) and three rectangular faces. Find the total area of just the two triangular faces.
Part C — Scale drawings
  1. A map has a scale of 1:25,000. Two towns are 5.6 cm apart on the map. Find the real distance in km.
  2. A scale drawing has scale 1:20. A model car is 22 cm long on the drawing. Find the real car’s length in metres.
Part D — Bearings
  1. Write 5° as a three-figure bearing.
  2. The bearing of M from N is 310°. Find the bearing of N from M.
  3. The bearing of R from S is 062°. Find the bearing of S from R.
  4. Point T is due south of point U. State the bearing of T from U.
Part E — Applied / challenge
  1. A ship sails on a bearing of 130° for 40 km, then changes course and sails on a bearing of 220° for 30 km. Describe (in words) how you’d construct a scale drawing to find the ship’s final position relative to its start.
  2. A scale drawing of a park uses a scale of 1:1500. If the actual park has an area of 90,000 m², find its area on the drawing, in cm².
  3. Explain why three-figure bearings are used instead of compass directions like “north-east” in navigation and surveying.

Key Vocabulary Recap

Construction — an accurate drawing made using only a ruler and compasses
Net — a 2D shape that folds into a 3D solid
Scale — the ratio between a drawing’s size and the real object’s size
Bearing — a direction, measured clockwise from North, written with three digits
Back bearing — the bearing of the reverse direction (± 180°)

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) sections C4.2 and C4.3 (Core subject content) and sections E4.2 and E4.3 (Extended subject content) — Geometrical constructions, and Scale drawings. As with Ratio, Proportion and Rates in the Number unit, the wording of these sections is identical for both tiers. Perpendicular bisector and angle bisector constructions are explicitly not required at either tier.

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