Area and Perimeter

Lengths, Areas and Volumes

The four shapes

Area is the space inside a shape, measured in squared units such as mm², cm² or m²

Shape	Area
Rectangle	length × width
Triangle	½ × base × perpendicular height
Parallelogram	base × perpendicular height
Trapezium	½ × (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, which is the straight-line distance between the two parallel edges rather than the length of 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 each case — 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

Exam tip

Use the perpendicular height, never a sloping side. On an obtuse triangle that height falls outside the triangle altogether, and mark schemes set aside a special case for candidates who use the sloping edge instead.

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
Multiply the total by whatever area one square represents, since one square is often not one unit