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C4.2 Geometrical constructions

Syllabus
0580–2028–2029
Topic
C4.2
Level
Core

Learning objectives

C4.2.1Measure and draw lines and angles• Measure and draw lines and angles using a ruler for every straight edge; perpendicular- and angle-bisector constructions are not required.C4.2.2Construct triangles• Construct a triangle from the lengths of all three sides using only a ruler and compasses; construction arcs must be shown.C4.2.3Draw and interpret nets• Draw, use and interpret nets of cubes, cuboids, prisms and pyramids, including using measurements from nets to calculate volumes and surface areas.

Measure and draw lines and angles accurately

Accurate geometry starts by placing the measuring scale on the feature itself: a ruler follows a line, while a protractor is centred on an angle's vertex.

Task Method Accuracy check
measure a line align ruler zero with one endpoint and read the other state the correct length unit
draw a line mark both endpoints at the required separation, then join with a ruler every straight edge is ruled
measure an angle centre the protractor at the vertex and align its baseline with one arm read the scale that begins at 0∘0^\circ on that arm
draw an angle draw one arm, mark the required degree position, then rule the second arm classify it as acute, right, obtuse or reflex

Perpendicular lines meet at 90∘90^\circ. When a task asks for a perpendicular line, use the grid or a protractor/set square to fix the right angle, then rule the line.

A correct-looking freehand edge is not an accurate construction. Also do not add compass constructions for perpendicular bisectors or angle bisectors: those constructions are outside this Core requirement.

Construct a triangle from three side lengths

Three side lengths fix a triangle by locating the third vertex at the intersection of two compass arcs.

  1. Rule one given side as the base.
  2. Set the compasses to the second side length and draw an arc from one endpoint.
  3. Set them to the third side length and draw an arc from the other endpoint.
  4. Mark the arc intersection as the third vertex and rule the two remaining sides.

Every point on the first arc is the second length from its centre, and every point on the second arc is the third length from its centre. Their intersection therefore satisfies both required distances.

For sides 6.56.5 cm, 6.56.5 cm and 88 cm, draw an 88 cm base. Two arcs of radius 6.56.5 cm—one from each endpoint—meet at the third vertex. Joining that point to both endpoints completes the triangle.

Leave both construction arcs visible and use only ruler and compasses. If the two shorter lengths do not add to more than the longest length, the arcs cannot form a non-degenerate triangle.

Draw, fold and interpret nets

A net is a flat arrangement of every face of a solid, joined along edges so that it can fold without overlap.

Solid Faces required in its net
cube six equal squares
cuboid three matching pairs of rectangles
triangular prism two matching triangles and three rectangles
square-based pyramid one square and four triangles

Start with one face, attach neighbouring faces along edges of equal length, and account for every face once. Use a ruler for all straight edges. Then mentally fold around each shared edge: faces must meet to close the solid, not cover the same position.

Dimensions transfer across matching edges. A cuboid net for dimensions 33 cm by 22 cm by 11 cm must contain two 3×23\times2, two 3×13\times1 and two 2×12\times1 rectangles. Their total area is the cuboid's surface area, 2(3×2+3×1+2×1)=222(3\times2+3\times1+2\times1)=22 cm2^2, and the enclosed volume is 3×2×1=63\times2\times1=6 cm3^3.

Having the right six faces is not sufficient if they overlap when folded or are joined along unequal edges. A net shows surfaces; hidden internal diagonals or duplicated faces are not included.