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Cambridge IGCSE0580

Area and perimeter

Mathematics 0580 Chapter Notes

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Area and perimeter
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In this chapter

  1. 1Rectangle: perimeter and area
  2. 2Triangle: area
  3. 3Parallelogram: area
  4. 4Trapezium: area
  5. 5Perimeter of a trapezium

1. Rectangle: perimeter and area

The perimeter is the total distance round the edge of a shape. The area is the amount of surface inside it.

For a rectangle, add all four sides for the perimeter. Multiply length by width for the area.

Perimeter is in cm, m or similar. Area is in square units, such as cm².

P = 2(l + w)

A = l × w

Worked example 12 marks

A rectangle has width 6 cm. The width of the rectangle is 1/12 of the perimeter of the rectangle. Work out the length of the rectangle.

  1. 1

    perimeter = 6 × 12 = 72 cm [1]

  2. 2

    l + w = 72 ÷ 2 = 36, so l = 36 - 6

  3. 3

    = 30 cm [1]

2. Triangle: area

The area of a triangle is half base times perpendicular height. This formula is given in the paper.

The height must meet the base at a right angle. It is not a sloping side.

If the question gives the area, put it into the formula and find the height (or the base) first.

A = 1/2 × b × h

Examiner insight

“Some did not appreciate that the first stage was to use the given area to find the height of the triangle.”

Worked example 11 mark

A triangle has base 10 cm and perpendicular height 6 cm. Work out the area of the triangle.

  1. 1

    A = 1/2 × 10 × 6

  2. 2

    = 30 cm² [1]

3. Parallelogram: area

The area of a parallelogram is base times perpendicular height.

The perpendicular height is measured at right angles to the base. A sloping side is not the height.

The formula is not given, so learn it.

A = b × h

Worked example 12 marks

Do not use a calculator. A parallelogram has base 9 cm and perpendicular height 4 cm. Its sloping side is 5 cm. Work out the area of the parallelogram.

  1. 1

    A = b × h = 9 × 4 [1]

  2. 2

    = 36 cm² (the 5 cm is not used) [1]

4. Trapezium: area

A trapezium has one pair of parallel sides. Its area is half the sum of the parallel sides times the perpendicular distance between them.

Add the two parallel sides, halve the total, then multiply by the perpendicular distance.

The formula is not given, so learn it. Give the units: an area in metres is in m².

A = 1/2(a + b)h

Examiner insight

“A variety of incorrect formulas were used.” “Some attempted to split the given shape into triangles and a rectangle.”

Worked example 13 marks

A trapezium has parallel sides 6 m and 20 m. The perpendicular distance between them is 10 m. One sloping side is 15 m. Work out the area of the trapezium. Give the units of your answer.

  1. 1

    A = 1/2(6 + 20) × 10 (the 15 m is not used) [1]

  2. 2

    = 13 × 10 = 130 [1]

  3. 3

    units: m² [1]

5. Perimeter of a trapezium

To find a perimeter, add all four sides. The sloping side is part of the perimeter, but it may not be given.

Split off a right-angled triangle. Pythagoras' theorem gives the sloping side. Do not find the area by mistake.

Examiner insight

“A few just added the three given sides and so a common incorrect answer was 44.” “Some calculated the area of the trapezium.”

Worked example 13 marks

A trapezium has parallel sides 21 cm and 15 cm. The side between them, at right angles to both, is 8 cm. Work out the perimeter of the trapezium.

  1. 1

    split off a right-angled triangle with shorter sides 8 cm and 21 - 15 = 6 cm [1]

  2. 2

    sloping side = √(8² + 6²) = √(100) = 10 cm [1]

  3. 3

    perimeter = 21 + 8 + 15 + 10 = 54 cm [1]

Recap

  • Perimeter is the distance round the edge. Area is in square units.
  • Rectangle: P = 2(l + w) and A = lw.
  • Triangle: A = 1/2bh, with the perpendicular height. This formula is given.
  • Parallelogram: A = bh. Use the perpendicular height, not the sloping side.
  • Trapezium: A = 1/2(a + b)h, where a and b are the parallel sides and h is the perpendicular distance between them.
  • Trapezium perimeter: add all four sides, including the sloping one.
  • Given an area, put it into the formula and solve for the missing length.

End-of-chapter exercise

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

  1. Do not use a calculator. A rectangle is 14 cm long and 9 cm wide. Work out its perimeter.2 marks
  2. A parallelogram has base 8.5 cm and perpendicular height 6 cm. Its sloping side is 7 cm. Work out its area.2 marks
  3. A trapezium has parallel sides 7.4 cm and 12.6 cm. The perpendicular distance between them is 5 cm. Work out its area.3 marks
  4. Do not use a calculator. A rectangle has area 96 cm² and width 8 cm. Work out its perimeter.3 marks
  5. A rectangle measures 2.5 m by 40 cm. Work out its area in cm².3 marks
  6. Do not use a calculator. A triangle has area 54 cm² and base 12 cm. Work out its perpendicular height.3 marks
  7. A garden is a rectangle 18 m by 10 m. A triangular flower bed with base 8 m and perpendicular height 5 m is dug in it. The rest is lawn. Lawn seed costs $3.50 per m². Work out the cost of seeding the lawn.4 marks
  8. Do not use a calculator. A trapezium has area 84 cm². Its parallel sides are 9 cm and 15 cm. Work out the perpendicular distance between them.3 marks
  9. Do not use a calculator. A parallelogram has perimeter 48 cm. Two of its sides are 15 cm long. The perpendicular distance between these two sides is 6 cm. Work out the perpendicular distance between the other two sides.4 marks
  10. A plot of land is a trapezium with area 480 m². Its parallel sides are 24 m and 36 m. A third side is at right angles to both parallel sides. The fourth side is 20 m. A fence is put round the whole plot. Fencing costs $13.75 per metre. Work out the cost of the fence.5 marks

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