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Cambridge IGCSE0580

Geometrical constructions

Mathematics 0580 Chapter Notes

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Geometrical constructions
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Geometrical constructions notes

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In this chapter

  1. 1Measuring and drawing lines and angles
  2. 2Constructing a triangle from three sides
  3. 3Equal sides and the rhombus
  4. 4Drawing nets
  5. 5Surface area and volume from a net

1. Measuring and drawing lines and angles

Use a ruler for every straight edge and a pencil.

To measure or draw an angle, put the centre of the protractor on the vertex and its baseline along one arm. Start counting from 0° on that arm. Two scales are printed, so check that your answer is sensible: an acute angle is below 90°.

A length that is a percentage longer is found with a multiplier. Do not stop at the increase.

60% longer = length × 1.6

Examiner insight

“The most common error involved reading the wrong scale on a 180° protractor, with 60° a common answer.” “Candidates should use a ruler and pencil and not go over in pen afterwards.”

Worked example 13 marks

Do not use a calculator. Line AB is 6 cm long. Draw a line AC that is 25% longer than AB and makes an angle of 40° with AB.

  1. 1

    25% of 6 = 1.5, so AC = 6 + 1.5 = 7.5 cm [1]

  2. 2

    protractor centre on A, baseline on AB, mark 40° from the 0° on AB [1]

  3. 3

    ruled line from A through the mark, 7.5 cm long [1]

2. Constructing a triangle from three sides

Draw the base with a ruler. Set the pair of compasses to the length of the second side. Draw an arc from one end of the base.

Reset to the length of the third side. Draw an arc from the other end. The arcs cross at the third vertex.

Leave in your construction arcs. They are the evidence that you constructed the triangle.

Examiner insight

“Common errors included omission of the required arcs, drawing an isosceles triangle with one side of the correct length, inaccuracies in arcs/lengths, and a half-scale triangle.” “The question asked candidates to leave in their construction arcs but a few tried to erase them or had arcs that were too small to see.”

Worked example 12 marks

In triangle ABC, AC = 10 cm and BC = 8.5 cm. Using a ruler and compasses only, construct triangle ABC. The line AB has been drawn for you.

  1. 1

    compasses set to 10 cm: arc from A. Compasses set to 8.5 cm: arc from B. Leave both arcs [1]

  2. 2

    the arcs cross at C. Join C to A and to B with a ruler: correct triangle [1]

3. Equal sides and the rhombus

In an isosceles triangle two sides are equal. An equilateral triangle has all three sides equal. Use the same compass setting for equal sides.

When an isosceles triangle is given only two lengths, test both ways round. Sides of 4 cm and 6 cm give 4 cm, 4 cm, 6 cm and 4 cm, 6 cm, 6 cm. These two triangles are not congruent.

A rhombus has four equal sides. Construct it by drawing two triangles on a shared diagonal.

Examiner insight

“It was quite rare to see both isosceles triangles since many simply repeated the first one on the second 4 cm line.” “An equilateral triangle with sides of 4 cm was often seen.”

Worked example 14 marks

An isosceles triangle has a side length 4 cm and a side length 6 cm. There are two possible triangles that can be drawn that are not congruent to each other. Construct these two triangles. One 4 cm side has been drawn for each of the triangles.

  1. 1

    triangle 1, sides 4, 4, 6: arc of radius 4 cm from one end and arc of radius 6 cm from the other [1]

  2. 2

    arcs cross: join with a ruler to give an accurate 4 cm, 4 cm, 6 cm triangle [1]

  3. 3

    triangle 2, sides 4, 6, 6: arcs of radius 6 cm from both ends of the 4 cm side [1]

  4. 4

    arcs cross: join with a ruler to give an accurate 4 cm, 6 cm, 6 cm triangle [1]

4. Drawing nets

A net is a flat shape that folds up into a solid. It has one face for every face of the solid, joined along edges that meet when it is folded.

A cube or cuboid has six faces. A triangular prism has three rectangles and two triangles. Draw every face, with a ruler, and include the inner lines between faces.

A net is flat. Do not draw a 3-D sketch.

Examiner insight

“Some candidates drew a net with only five faces, representing an open box.” “A significant number appeared unfamiliar with drawing a net leading to no attempt or a 3-D drawing.”

Worked example 13 marks

Draw an accurate net of a cuboid 4 cm by 3 cm by 2 cm.

  1. 1

    six faces: two 4 cm by 3 cm, two 4 cm by 2 cm, two 3 cm by 2 cm [1]

  2. 2

    a row of four rectangles 2 cm high, with widths 4 cm, 3 cm, 4 cm, 3 cm [1]

  3. 3

    a 4 cm by 3 cm rectangle above and one below the first 4 cm wide face, all ruled with inner lines [1]

5. Surface area and volume from a net

The surface area of a solid is the total area of all the faces in its net. Find the area of each face, then add.

A cuboid has three different sizes of rectangle. Each size appears twice.

Volume comes from the lengths on the net. Give square units for area and cubic units for volume.

surface area = sum of the areas of all the faces

volume of a cuboid = length × width × height

Examiner insight

“Some did not realise there were 3 different size rectangles, repeating for example 10 × 5 while others who knew exactly what to do wrote 2(10 × 4) + (10 × 5) + (5 × 4) = 150.” “The main error was to work out the volume in part (a) and then often the surface area in part (b).”

Worked example 14 marks

Do not use a calculator. The net of a cuboid is made from pairs of rectangles measuring 10 cm by 5 cm, 10 cm by 4 cm and 5 cm by 4 cm. Work out the surface area and the volume of the cuboid.

  1. 1

    face areas: 10 × 5 = 50, 10 × 4 = 40, 5 × 4 = 20 [1]

  2. 2

    surface area = 2 × (50 + 40 + 20) = 220 cm² [1]

  3. 3

    volume = 10 × 5 × 4 [1]

  4. 4

    = 200 cm³ [1]

Recap

  • Use a ruler for every straight edge. Start the protractor from 0° on the arm.
  • A length 60% longer is the original length × 1.6.
  • Construct a triangle with a pair of compasses. Leave in the construction arcs.
  • Do not measure when the question says compasses only.
  • A rhombus is two triangles on a shared diagonal, with the same arc setting.
  • A net is flat, has every face, and has every inner line.
  • Surface area is the sum of all the face areas. Check you have used each size twice.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. A line is 5 cm long. A second line is 60% longer. Find the length of the second line.2 marks
  2. Do not use a calculator. A protractor has two scales. A student measures an acute angle. One scale reads 35 and the other reads 145. State the size of the angle.1 mark
  3. Describe how to draw an angle of 65° at the point P on the line PQ.3 marks
  4. The line AB is 6 cm long. Describe how to construct triangle ABC with AC = 5 cm and BC = 4 cm, using a ruler and compasses only.3 marks
  5. A rhombus ABCD has sides of 5 cm. The diagonal AC is 8 cm. Describe how to construct the rhombus using two triangles.4 marks
  6. State the number of faces on the net of a triangular prism, and name their shapes.2 marks
  7. Do not use a calculator. A cube has a volume of 27 cm³. Work out the total area of the net of the cube.3 marks
  8. Do not use a calculator. A gift box is a cuboid with a square base of side 6 cm. Its volume is 180 cm³. Work out the height of the box. Then work out the area of card needed for the net.4 marks
  9. The net of a prism is made from two triangles and three rectangles. Each triangle has sides 5 cm, 5 cm and 6 cm, and its height measured from the 6 cm side is 4 cm. Each rectangle is 15 cm long and its width is one side of the triangle. Work out the total surface area and the volume of the prism.5 marks
  10. Do not use a calculator. A rectangular sheet of card measures 26 cm by 20 cm. A square of side 4 cm is cut from each of its four corners. The shape left is the net of an open tray, and its four sides are folded up. Work out the volume of the tray. Then work out the area of card needed for the net of a closed cuboid of the same size.6 marks

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