CAMBRIDGE O LEVEL · 2025–2027
Geometrical constructions.
Measure accurately, construct with ruler and compasses, and unfold solids into nets.
BY THE END
Measure and draw lines and angles · construct SSS triangles with visible arcs · interpret nets and their measurements
01
Measure accurately, then draw deliberately
Accurate geometry depends on using tools with a clear starting point. A ruler measures from its zero mark, and a protractor measures a turn from a chosen baseline through the vertex.
Place the tool correctly before reading a scale. With a protractor, align the centre with the vertex and choose the scale that starts at zero on the baseline. Estimate whether the angle is acute or obtuse first, so an incorrect scale reading is obvious.
Use a ruler for every straight edge. Align a protractor’s centre with the vertex and its baseline with one arm; read from the scale that starts at 0° on that arm.
WORKED EXAMPLE
Draw a 65° angle with a 7 cm arm.
- Draw a 7 cm line segment and label its endpoints.
- Centre the protractor at the chosen vertex and align its baseline.
- Mark 65° on the correct scale.
- Use a ruler to draw a ray from the vertex through the mark.
Pause and explainWhy should you not automatically start measuring at the ruler’s physical edge?
The printed zero mark may be inset. Measurements must begin from the scale’s zero, not the edge of the tool.
02
Construct a triangle from three side lengths
Constructing a triangle means finding a vertex that satisfies two or more distance or angle conditions at the same time. A compass arc records all points at a fixed distance from its centre.
Start with a known side, then use the remaining information to locate the third vertex. For three known sides, intersect two arcs. For known angles, draw the required rays. Keep construction arcs visible so the method can be checked.
RULER AND COMPASSES · BUILD A 4–5–6 TRIANGLE
Draw AB = 6 cm using a ruler.
Schematic on screen: use the stated centimetre measurements on paper.
The two arcs find a point 5 cm from A and 4 cm from B at the same time. A reflected triangle below AB is also possible.
Before constructing, check the two shorter sides add to more than the longest side. Lengths 2 cm, 3 cm and 6 cm cannot form a triangle.
Pause and explainWhy do intersecting compass arcs locate a triangle vertex?
An intersection is at the required distance from each arc’s centre, so it satisfies both length conditions simultaneously.
03
Read a net as connected faces
A net must preserve the lengths and arrangements of the solid’s surfaces. Faces that share an edge when folded must have matching edge lengths on the net.
Label a rough plan first, then draw to the stated measurements. Check fold relationships as well as individual dimensions: correctly sized faces in an impossible arrangement do not form the intended solid. Use a ruler for every required straight edge.
Cuboid
Triangular prism
Square-based pyramid
When folded, matching edges must have equal lengths, all faces must be present, and the faces must not overlap. Six squares do not automatically make a valid cube net.
WORKED EXAMPLE
A cuboid net has dimensions 4 cm, 3 cm and 2 cm. Find surface area and volume.
The faces come in three pairs: 4 × 3, 4 × 2 and 3 × 2.
Surface area = 2(12 + 8 + 6) = 52 cm²
Volume = 4 × 3 × 2 = 24 cm³
Volume = 4 × 3 × 2 = 24 cm³
WORKED EXAMPLE
A triangular-prism net has two 3–4–5 right triangles and a prism length of 6 cm.
Each triangular end has area 12 × 3 × 4 = 6 cm². Three rectangles have total area 6(3 + 4 + 5) = 72 cm².
Surface area = 2 × 6 + 72 = 84 cm²
Volume = 6 × 6 = 36 cm³
Volume = 6 × 6 = 36 cm³
Pause and explainWhat should you check when two faces will share an edge?
Their joining edge lengths must match, and their positions must allow folding without overlapping faces.
04
Your turn. Measure and fold mentally.
Identify the given measurements and plan the order of construction before drawing. Estimate the final shape so you can detect an impossible angle or length early.
Use sharp, clear marks and retain arcs where they show the construction. Measure the completed figure against the original conditions rather than assuming a neat drawing must be correct.
1
Describe a ruler-and-compasses construction for a triangle with sides 5 cm, 6 cm and 7 cm.
Reveal worked answer
Draw a 7 cm base. Draw an arc of radius 5 cm from one end and radius 6 cm from the other. Join an intersection to both endpoints with a ruler; keep the arcs.
2
How many faces, and of what shapes, appear in a square-based pyramid net?
Reveal worked answer
Five faces: one square base and four triangles.
3
A cube net contains six squares of side 3 cm. Find its total area and the cube’s volume.
Reveal worked answer
Total area = 6 × 3² = 54 cm². Volume = 3³ = 27 cm³.