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Cambridge IGCSE0607

Area and perimeter

International Mathematics 0607 Chapter Notes

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Area and perimeter
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In this chapter

  1. 1Perimeter and compound shapes
  2. 2Rectangle and parallelogram
  3. 3Triangle
  4. 4Trapezium
  5. 5Area and perimeter together
  6. 6Finding a missing length from an area

1. Perimeter and compound shapes

The perimeter is the distance all the way round the outside of a shape. Add the lengths of the outer edges only. Do not add any lines inside the shape.

For a rectangle, opposite sides are equal. Perimeter is measured in cm, not cm².

In an L-shape or a staircase shape, the edges that run across add up to the full width and the edges that run up add up to the full height. This is quicker than adding every edge.

P = 2(l + w)

Examiner insight

“The quick method is to realise that the tops of the rectangles add to 12 and the widths on the right-hand side add to 8, so the perimeter is 2(12 + 8) = 40 cm.”

Worked example 12 marks

A shape is made by cutting a rectangle 4 cm by 3 cm out of one corner of a rectangle 10 cm by 8 cm. All the corners are right angles. Work out the perimeter of the shape.

  1. 1

    the cut-out only moves two edges inwards, so the edges that run across add to 10 cm and the edges that run up add to 8 cm [1]

  2. 2

    P = 2(10 + 8) = 36 cm [1]

2. Rectangle and parallelogram

The area of a rectangle is length × width.

The area of a parallelogram is base × perpendicular height. The perpendicular height meets the base at a right angle. It is not the sloping side.

The sloping side is used for the perimeter, because opposite sides are equal.

A = l × w

A = b × h

Examiner insight

“Quite a number of candidates worked out the length of BC or AD using Pythagoras’ theorem and then multiplied their answer by 6 as if the shape was a rectangle.”

Worked example 14 marks

A parallelogram has a base of 9 cm, a perpendicular height of 4 cm and a sloping side of 5 cm.

(a) Work out the area.

(b) Work out the perimeter.

  1. 1

    (a) area = base × perpendicular height [1]

  2. 2

    = 9 × 4 = 36 cm² [1]

  3. 3

    (b) opposite sides are equal: P = 2(9 + 5) [1]

  4. 4

    = 28 cm [1]

3. Triangle

The area of a triangle is half the base multiplied by the perpendicular height. It is the only formula in this topic that is given.

The base and the height must meet at a right angle. In a triangle with a sloping side, the sloping side is not the height.

The perimeter of a triangle is the sum of its three sides.

A = 1/2 × b × h

Examiner insight

“Although the majority of candidates found the answer correctly, some forgot to divide by 2 to find the area of the triangle.” “The lowest achieving often did not know how to find the area of a triangle; errors included finding perimeter, multiplying base and height without dividing by two, and multiplying the lengths of two or three of the sides.”

Worked example 12 marks

A triangle has a base of 12 cm, a sloping side of 13 cm and a perpendicular height of 5 cm. Work out the area of the triangle.

  1. 1

    area = 1/2 × base × perpendicular height = 1/2 × 12 × 5 [1]

  2. 2

    = 30 cm² [1]

4. Trapezium

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

This formula is not given, so learn it. You can also split the trapezium into a rectangle and a triangle and add their areas.

A = 1/2(a + b)h

Examiner insight

“In part (i), most candidates realised that they could split the shape into a rectangle and a triangle to work out the area of the trapezium.” “A few knew the area of a trapezium formula and applied it correctly.”

Worked example 13 marks

A trapezium has parallel sides of 6 cm and 10 cm. The perpendicular distance between them is 4 cm. Work out the area of the trapezium.

  1. 1

    area = 1/2(a + b)h [1]

  2. 2

    = 1/2 × (6 + 10) × 4 [1]

  3. 3

    = 32 cm² [1]

5. Area and perimeter together

Area measures the surface inside a shape, in square units such as cm². Perimeter measures the distance round it, in units of length such as cm.

Find the lengths you need from the coordinates or the figure first. Then work out the perimeter and the area as two separate answers, each with its own unit.

Examiner insight

“Often, cm2 was given as the units to both area and perimeter. A few mixed up area and perimeter.” “The candidates could work out the area of the triangle but many did not give the correct units.”

Worked example 12 marks

Quadrilateral ABCD is drawn on a 1 cm square grid.

(c) Work out the perimeter and the area of quadrilateral ABCD.

  1. 1

    read the sides off the axes: AB = 2 - (-4) = 6 cm and AD = 3 - (-2) = 5 cm

  2. 2

    perimeter = 2(6 + 5) = 22 cm [1]

  3. 3

    area = 6 × 5 = 30 cm² [1]

6. Finding a missing length from an area

Sometimes the area is known and a length is missing. Write the formula, substitute what you know, then rearrange to find the missing length.

When two shapes have the same area, work out the area of the one you can first. Then put it equal to the area formula of the other.

Examiner insight

“This is the formula for the area of a triangle with the area, A, and the height, h, given so the length of the base, b, has to be found. So, this was not a straightforward substitution question as the equation has to be re-arranged first before the substitution.”

Worked example 13 marks

A triangle and a rectangle have the same area. The triangle has a base of 10 cm and a perpendicular height of 9 cm. The rectangle has a width of 6 cm and a length of x cm. Work out the value of x.

  1. 1

    area of triangle = 1/2 × 10 × 9 = 45 cm² [1]

  2. 2

    area of rectangle: 6x = 45 [1]

  3. 3

    x = 45 ÷ 6 = 7.5 [1]

Recap

  • Perimeter is the distance round the outside. Its unit is cm.
  • Area is the surface inside. Its unit is cm².
  • Rectangle: A = l × w. Parallelogram: A = b × h, with h at a right angle to b.
  • Triangle: A = ½ × b × h. This formula is given.
  • Trapezium: A = ½(a + b)h. Learn it, or split into a rectangle and a triangle.
  • To find a missing length, write the formula, substitute, then rearrange.

End-of-chapter exercise

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

  1. A rectangle is 7 cm by 5 cm. Work out its perimeter.2 marks
  2. Do not use a calculator. A rectangle has a length of 12 m and a width of 8 m. Work out its area. Give the unit.2 marks
  3. Do not use a calculator. A triangle has a base of 14 cm and a perpendicular height of 6 cm. Work out its area.2 marks
  4. A parallelogram has a base of 15 cm, a perpendicular height of 8 cm and a sloping side of 10 cm. Work out (a) the area (b) the perimeter.3 marks
  5. Do not use a calculator. A trapezium has parallel sides of 7 cm and 11 cm. The perpendicular distance between them is 6 cm. Work out its area.3 marks
  6. Do not use a calculator. The perimeter of a rectangular garden is 44 m. Its length is 14 m. Work out the area of the garden.3 marks
  7. A triangle has a base of 80 cm and a perpendicular height of 1.5 m. Work out its area in m².3 marks
  8. Do not use a calculator. A rectangle 12 cm by 9 cm has a rectangle 5 cm by 4 cm cut out of one corner. All the corners are right angles. Work out (a) the perimeter of the shape (b) the area of the shape.4 marks
  9. Do not use a calculator. A sign-maker cuts two badges from a sheet of metal. One badge is a parallelogram with a base of 16 cm, a sloping side of 6 cm and a perpendicular height of 4.5 cm. The other badge is a trapezium. Its parallel sides are 5 cm and x cm, and they are 8 cm apart. The two badges have the same area. Work out the value of x.4 marks
  10. The end wall of a barn is a trapezium. Its two parallel edges are vertical, with heights 2.4 m and 3.6 m, and they are 5 m apart. The wall has a rectangular window 150 cm by 80 cm and a triangular vent with a base of 60 cm and a perpendicular height of 40 cm. The rest of the wall is painted with two coats of paint. One tin of paint covers 4 m². Work out the number of tins needed.6 marks

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