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Constructions And Scale Drawings

Constructions And Scale Drawings

Constructing triangles and nets

Constructing Triangles Core + Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. Constructing a triangle from three sides
  4. Nets
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 4.1 and 4.2: constructions and nets
  • 4.2Measure and draw lines (nearest mm) and angles (nearest degree).
  • 4.2Construct a triangle from its three sides with a ruler and compasses only, leaving the arcs.
  • 4.2Draw and use nets of cubes, cuboids, prisms and pyramids for surface area and volume.

Next: Scale drawings and bearings.

Formula bank

Triangle from three sidesLearn itbase, then an arc from each end, join to the crossingleave all arcs.Use it when 'ruler and compasses only'.
Will it make a triangle?Learn itthe two shorter sides must add to MORE than the longest5, 6, 8 works; 3, 4, 9 doesn't.Use it when checking lengths.
NetLearn itsurface area of the solid = total area of its netUse it when nets and surface area.

Constructing a triangle from three sides

Draw AB

Constructing a triangle from three sidesAB9 cm
Rule AB = 9 cm.

Usually AB is drawn for you. Check it with your ruler.

Radius 6 cm

Constructing a triangle from three sidesAB9 cm
Compasses set to 6 cm, point on A.

Set the compasses against the ruler; draw an arc above AB.

Radius 7 cm

Constructing a triangle from three sidesAB9 cm
Compasses set to 7 cm, point on B.

Draw an arc crossing the first one. Do not rub out either arc.

C is the crossing

Constructing a triangle from three sides51°AB9 cmC67
Join AC and BC.

Join the crossing to A and B with a ruler. Angle BAC measures about 51°.

General methodHow to construct a triangle from three sides

  1. Draw (or check) the base with a ruler.
  2. Set the compasses to the second length against a ruler; arc from one end of the base.
  3. Set them to the third length; arc from the other end so the arcs cross.
  4. Join the crossing to both ends. LEAVE the arcs.
  5. If asked, measure an angle with a protractor (read the scale starting at 0 on that arm).
No arcs = no method marks. Use a sharp pencil: a 2 mm error can push a measured angle outside the allowed ±2°.

Try it yourself

1
Can sides 3 cm, 4 cm and 9 cm make a triangle? Type yes or no.
Hint
3 + 4 = 7.
Show answer

Answer no

2
Can sides 5 cm, 6 cm and 8 cm make a triangle? Type yes or no.
Show answer

Answer yes

Nets

Net of a 4 cm by 3 cm by 2 cm cuboid4×34×24×24×32×32×3
One net of a 4 cm × 3 cm × 2 cm cuboid: three pairs of rectangles.

A net is the flat shape that folds into the solid. Its area is the solid's surface area: \(2(12+8+6)=52\) cm². Volume \(=4\times3\times2=24\) cm³.

General methodHow to use a net

  1. Identify each face: a cuboid has 3 pairs of equal rectangles; a cube 6 equal squares; a prism 2 ends + rectangles.
  2. Surface area = sum of the face areas.
  3. Read the solid's dimensions from the net to find the volume.

Try it yourself

1
Surface area of a 5 cm cube (cm²)?
Show answer

Answer 150

2
Surface area of a 4 cm × 3 cm × 2 cm cuboid (cm²)?
Show answer

Answer 52

Ways they test it

Type 1 Construct and measure

"Using a ruler and compasses only, construct triangle ABC. AB has been drawn. Measure angle BAC." (3 marks)

Spot it 'ruler and compasses only'.

  1. Arcs from both ends.
  2. Join; measure (±2°).

Type 2 Can it be drawn?

"Explain why a triangle with sides 3 cm, 4 cm and 9 cm cannot be constructed." (1 mark)

Spot it 'explain why'.

  1. Compare the two shorter sides with the longest.

For example 3 + 4 = 7 < 9

    Answer the arcs don't meet

    Type 3 Nets

    "Draw a net of the cuboid" / "Find the surface area from the net." (2 marks)

    Spot it a net or a solid.

    1. Pairs of faces.
    2. Add the areas.

    Note: papers before 2025 also asked for perpendicular and angle bisectors with compasses; they are not in the current syllabus.

    Mistakes to avoid

    • ✗ Rubbing out the arcs✓ leave themThey earn the marks.
    • ✗ Compasses set by eye✓ set against a ruler±2 mm tolerance.
    • ✗ Reading the wrong protractor scale✓ start from 0 on the arm130° vs 50°.

    Summary

    • Triangle from 3 sides: base, two arcs, join; leave the arcs.
    • Shorter two sides must add to more than the longest.
    • A net's area = the solid's surface area.

    If you see …

    • "ruler and compasses only"arcs, never a protractor for the sides.
    • a netfind each face, then add the areas.

    Practice

    Practice: easy, medium, hard

    Easy

    1
    Which sides make a triangle: 1, 2, 3 or 5, 6, 8? Type the set that works, e.g. 5,6,8.
    Show answer

    Answer 5, 6, 8

    2
    How many faces does a cuboid's net have?
    Show answer

    Answer 6

    3
    Volume of a 4 cm × 3 cm × 2 cm cuboid (cm³)?
    Show answer

    Answer 24

    Medium

    1
    Triangle PQR: PQ = 8 cm, PR = 5 cm, QR = 6 cm. Angle QPR to the nearest degree (use the cosine rule to check a construction)?
    Guide me

    What is it asking? an angle

    What do we know? construct, then measure

    Plan should be about 48-49°

    Show working
    1. cosine rule gives 48.5°; a measurement of 47° to 51° is accepted

    Answer 49

    2
    Surface area of a cube of edge 3 cm (cm²)?
    Guide me

    What is it asking? an area

    What do we know? 6 equal squares

    Plan 6 × 9

    Show working
    1. 54

    Answer 54

    3
    A net has two 4 × 3, two 4 × 2 and two 3 × 2 rectangles. Surface area (cm²)?
    Guide me

    What is it asking? an area

    What do we know? pairs

    Plan 2(12 + 8 + 6)

    Show working
    1. 52

    Answer 52

    Hard

    1
    Triangle XYZ: XY = 10 cm, XZ = 7 cm, YZ = 8 cm. Angle XYZ to the nearest degree?
    Guide me

    What is it asking? an angle

    What do we know? construct and measure

    Plan check: about 44°

    Show working
    1. cosine rule: 44°

    Answer 44

    PartAnswer / workingMarksGuidance
    triangle with arcsB2
    44 ± 2B1
    Total: 3 marks
    2
    Can sides 2 cm, 2 cm and 5 cm make a triangle? Type yes or no.
    Guide me

    What is it asking? yes or no

    What do we know? 2 + 2 = 4

    Plan 4 < 5

    Show working
    1. no: the arcs don't meet

    Answer no

    PartAnswer / workingMarksGuidance
    no with reasonB1
    Total: 1 mark
    3
    A cube has surface area 96 cm². Find its edge (cm).
    Guide me

    What is it asking? a length

    What do we know? one face = 96 ÷ 6

    Plan √16

    Show working
    1. 16 cm² per face
    2. edge 4 cm

    Answer 4

    PartAnswer / workingMarksGuidance
    16M1
    4A1
    Total: 2 marks

    Can I do this?

    Anything unticked: revise that section first. All ticked: take the quiz.

    Quiz

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