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E4.2 Geometrical constructions

Syllabus
0580–2028–2029
Topic
E4.2
Level
Extended

Learning objectives

E4.2.1Measure and draw lines and angles.• Measure and draw lines and angles.E4.2.2Construct a triangle, given the lengths of all sides, using a ruler and• Construct a triangle, given the lengths of all sides, using a ruler and pair of compasses only.E4.2.3Draw, use and interpret nets• Draw, use and interpret nets. A ruler should be used for all straight edges. Constructions of perpendicular bisectors and angle bisectors are not required. e.g. construct a rhombus by drawing two triangles. Construction arcs must be shown. Examples include: • draw nets of cubes, cuboids, prisms and pyramids • use measurements from nets to calculate volumes and surface areas.

Measure and draw accurately

Accurate geometry starts from the correct scale: place the ruler or protractor at the true starting point, read the required unit, and draw every straight edge with a ruler.

Job Set-up Read or mark
measure a line align the ruler's zero with one endpoint read the other endpoint in the requested unit
draw a line mark both endpoints at the required separation join them with one ruled edge
measure an angle centre the protractor on the vertex and align its baseline with one arm use the scale that begins at 0∘0^\circ on that arm
draw an angle draw one arm, mark the required degree value, then rule the second arm keep the protractor centre fixed on the vertex

For a reflex angle, first measure the smaller angle between the arms, then subtract it from 360∘360^\circ. For example, if the smaller angle is 137∘137^\circ, the reflex angle is 360∘−137∘=223∘360^\circ-137^\circ=223^\circ.

Check that the line begins at zero rather than at the ruler's physical edge, that centimetres and millimetres have not been confused, and that the chosen protractor scale matches the angle's visible size.

Construct a triangle from three side lengths

A three-side triangle construction locates the third vertex as the intersection of two distance arcs. Every point on an arc is the same distance from its centre.

  1. Draw one given side accurately with a ruler.
  2. Put the compass point at one endpoint and set its width to the second side length; draw an arc.
  3. Without estimating, reset the compass to the third side length, centre it at the other endpoint and draw an intersecting arc.
  4. Use a ruler to join the arc intersection to both endpoints.
  5. Leave both construction arcs visible.

For sides 55 cm, 88 cm and 1010 cm, draw the 1010 cm base. From one endpoint draw an arc of radius 55 cm and from the other an arc of radius 88 cm. Their intersection is the third vertex; joining it to the base endpoints fixes all three lengths.

A ruler measures and joins; it must not be used to guess the third vertex. Two clear intersecting arcs are part of the construction evidence. If the chosen side lengths cannot make the arcs intersect, they cannot form a triangle.

Draw, fold and use a net

A net is a flat arrangement of every face of a solid, joined along edges so it can fold without gaps or overlapping faces.

Solid Faces the net must contain
cube 6 equal squares
cuboid 3 matching pairs of rectangles
triangular prism 2 matching triangles and 3 rectangles
square-based pyramid 1 square and 4 triangles

Draw all straight edges with a ruler and preserve every face dimension. To interpret a net, choose one face as the base, imagine adjacent faces folding through 90∘90^\circ, and track which free edges and labelled vertices meet. To draw a net, arrange the complete face set so no two faces would occupy the same position after folding.

\text{surface area}=\sum \text{area of every face},\qquad \text{volume}=\text{cross-sectional area}\times\text{perpendicular length}

A cuboid measuring 66 cm by 33 cm by 22 cm needs rectangles in three matching pairs: 6×36\times3, 6×26\times2 and 3×23\times2. Its surface area is 2(18+12+6)=722(18+12+6)=72 cm2^2, and its volume is 6×3×2=366\times3\times2=36 cm3^3.

Correct face sizes alone do not guarantee a valid net: their connections must also fold correctly. Surface area uses all faces and square units; volume uses enclosed three-dimensional space and cubic units. Perpendicular-bisector and angle-bisector constructions are not required here.