MathematicsExam code: 0580

Area and Perimeter

Lengths, Areas and Volumes

PractiseFlashcards

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

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

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