0580

Area and Perimeter

Lengths, Areas and Volumes

Adding or subtracting

  • A compound shape is built from simpler ones, and there are two ways to handle it
    • Split it into rectangles and triangles, find each area, and add them
    • Subtract, by taking the whole enclosing rectangle and removing the missing piece
  • Subtracting is usually quicker where a corner or a hole has been cut out
A shape made by slicing a corner off a rectangle, with the missing corner drawn back in as a dashed triangle so the area can be found as the full rectangle minus that triangle
Source: Adding & subtracting areas by Save My Exams
  • Sketch the split on the diagram and label each piece, so the working shows where each number came from
A compound shape with right angles marked at three corners, 4 cm up the left side, 5 cm along a short top edge, then a sloping edge rising to a right side of 9 cm, over a base of 12 cm
Source: Adding & subtracting areas by Save My Exams
  • Check the split covers the shape exactly once, with no overlap and no gap
The compound shape split into a rectangle and a triangle by a dashed line, with the two lengths the split needs worked out on the diagram: 12 − 5 = 7 cm along the dashed line and 9 − 4 = 5 cm up the right-hand side
Source: Adding & subtracting areas by Save My Exams
Exam tip

Label each piece you split the shape into and write its area on its own line. Those separate areas are where the method marks sit, so a shape answered in one unexplained number scores nothing if the total is wrong.

Worked example

The same area two ways

The floor plan of the same L-shaped room is split into two rectangles by the dashed line. Work out the floor area.

The L-shaped floor plan divided by a dashed vertical line into Area A on the left and Area B on the right, with 3.6 m across the top of Area A, 3.2 m up the left side, 2.4 m across the top of Area B, 1.8 m up the right side and 6 m along the bottom
Source: Problem-solving with areas by Save My Exams

Solution:

  • Area A is a rectangle 3.6 m by 3.2 m, so its area is 3.6 × 3.2 = 11.52 m²
  • Area B is a rectangle 2.4 m by 1.8 m, so its area is 2.4 × 1.8 = 4.32 m²
  • Adding: 11.52 + 4.32 = 15.84 m²
  • Subtracting gives the same answer, which is a quick check
  • The whole enclosing rectangle is 6 × 3.2 = 19.2 m²
  • The piece missing from the top right is 2.4 m by 3.2 − 1.8 = 1.4 m, so 2.4 × 1.4 = 3.36 m²
  • 19.2 − 3.36 = 15.84 m², as before

Working backwards

  • A question may give the area and ask for a missing length, which means substituting into the formula and solving
  • The same applies to perimeter, and a shape with an algebraic side turns into an equation
  • Check the units the answer needs, converting only at the end
Worked example

An area given, a length wanted

A trapezium has parallel sides of 9 cm and 13 cm and an area of 77 cm². Find the distance between the parallel sides.

A trapezium with its two parallel sides horizontal: the shorter top side is labelled 9 cm and the longer bottom side 13 cm. A dashed line runs straight down from the top left corner to the bottom side, meeting it at a marked right angle, and carries no label.

Solution:

  • Write the formula first, then substitute what is known
  • Area = ½ × (sum of the parallel sides) × distance between them
  • 77 = ½ × (9 + 13) × h
  • 77 = ½ × 22 × h
  • 77 = 11h
  • h = 77 ÷ 11 = 7 cm
  • Check by substituting back: ½ × 22 × 7 = 77, which matches the area given
  • The answer is a length, so it is in cm and not cm²