Mathematics (Core)Exam code: 0580

Area and Perimeter

Lengths, Areas and Volumes

PractiseFlashcards

The four shapes

  • Area is the space inside a shape, measured in squared units such as mm², cm² or m²
ShapeArea
Rectanglelength × width
Triangle½ × base × perpendicular height
Parallelogrambase × perpendicular height
Trapezium½ × (sum of the parallel sides) × distance between them
  • Only the triangle formula is printed in the paper, so the other three have to be known
  • Every one of them uses the perpendicular height, measured at right angles to the base (up to the opposite corner of a triangle, or across to the opposite parallel side) rather than along a sloping side
The four area formulas with a labelled diagram for each: a rectangle of length l and width w with area l × w, a triangle of base b and perpendicular height h with area a half b h, a trapezium with parallel sides a and b and height h with area a half of a plus b, all times h, and a parallelogram of base b and perpendicular height h with area b h, the height drawn as a dashed line meeting the base at a right angle in the triangle, the trapezium and the parallelogram
Source: Area Formulas by Save My Exams
  • The trapezium formula needs the two parallel sides identified first, since the other two play no part
  • If a formula escapes you, cut the shape into a rectangle plus one or two triangles instead and add those areas: the total is the same

Worked example

Three areas from the formulas

Find the area of each shape.

A right-angled triangle with a horizontal side of 8 cm and a vertical side of 7 cm meeting at the marked right angle, so those two sides are the base and the perpendicular height
Source: Area Formulas by Save My Exams
A parallelogram with a base of 15 cm and a perpendicular height of 12 cm marked by a double-headed arrow drawn straight down the side rather than along the sloping edge
Source: Area Formulas by Save My Exams
A trapezium with parallel sides of 15 cm along the top and 30 cm along the bottom, and a perpendicular height of 20 cm marked by a double-headed arrow between them
Source: Area Formulas by Save My Exams

Solution:

  • The triangle is right-angled, so the two sides meeting at the right angle are the base and the perpendicular height
  • Triangle: ½ × 8 × 7 = 28 cm²
  • On the parallelogram the 12 cm is the perpendicular height, not the sloping side
  • Parallelogram: 15 × 12 = 180 cm²
  • On the trapezium the parallel sides are 15 cm and 30 cm, and they are 20 cm apart
  • Trapezium: ½ × (15 + 30) × 20 = ½ × 45 × 20 = 450 cm²
  • Every answer is in cm² because area is measured in squared units

Areas on a grid

  • Count the whole squares inside the shape first, marking them off as you go
  • Then pair up the part-squares round the edge to make whole ones
  • Check what one square stands for: on a 1 cm² grid the count is the area in cm², and if a square stands for any other area, multiply the count by that area

Worked example

Areas on a 1 cm² grid

(a) The triangle on the first 1 cm² grid has its corners at (1, 1), (7, 1) and (3, 5). Find its area.

A coordinate grid from 0 to 9 across and 0 to 6 up, with a triangle ruled between the points (1, 1), (7, 1) and (3, 5); nothing else is marked

(b) Estimate the area of the shape with a curved side on the second 1 cm² grid.

A grid of 1 cm squares with a shape standing on its bottom edge: two straight vertical sides 5 squares apart, the left one about 2.3 squares tall and the right one about 1.3, and a curved top that rises from the left side, peaks about 4.6 squares up and falls to the right side; no squares are marked

Solution:

  • (a) The side from (1, 1) to (7, 1) lies along a grid line, so it makes a good base: 7 − 1 = 6 cm
  • The perpendicular height runs straight up from that base to the corner at (3, 5): 5 − 1 = 4 cm
  • Area = ½ × 6 × 4 = 12 cm²
  • With its corners on grid points the formula is exact, which is quicker and safer than counting the cut squares along the two sloping sides
  • (b) A curved side has no formula, so count instead: 13 squares lie wholly inside the shape
  • The curved side cuts through 10 more squares; these are roughly half covered each, so together they make about 10 ÷ 2 = 5 whole squares
  • Estimated area = 13 + 5 = 18 cm²
  • The part squares are only roughly halves, so this is an estimate, and a value a square or so either way is just as reasonable

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