Perimeter is the distance around a shape; area is the space inside it. Four formulas cover rectangles, triangles, parallelograms and trapezia, and only one of them is printed on the exam paper.
Area always uses a length and the PERPENDICULAR height: rectangle \(lw\), triangle \(\tfrac{1}{2}bh\), parallelogram \(bh\), trapezium \(\tfrac{1}{2}(a+b)h\). Only the triangle formula is in the List of formulas, so learn the other three. Perimeter is just the sum of the sides.
Fig. 1: a trapezium with parallel sides 6 cm and 10 cm, 4 cm apart (drawn to scale); the height is measured at right angles to the parallel sides
Section 01
Rectangles and triangles.
In the lab, change the size of the rectangle and watch the area and perimeter change. Then switch to a triangle and press Add a copy: two identical triangles make a parallelogram, which is why a triangle is HALF of base times height.
Shape
Area
Given in exam?
Rectangle
\(A = lw\)
No
Triangle
\(A = \tfrac{1}{2}bh\)
Yes (List of formulas)
Parallelogram
\(A = bh\)
No
Trapezium
\(A = \tfrac{1}{2}(a+b)h\)
No
Worked example Rectangle
A rectangle is 12 cm long and 7.5 cm wide. Find its area and its perimeter.
Area \(= l \times w = 12 \times 7.5 = 90\).
Perimeter = sum of all four sides \(= 12 + 7.5 + 12 + 7.5 = 39\).
Answer: Area \(90\text{ cm}^2\), perimeter 39 cm
Worked example Triangle with a sloping side
A triangle has base 10 cm. Its sloping side is 7 cm and its perpendicular height is 5 cm. Find its area.
The height in the formula must be at right angles to the base, so use 5 cm, not 7 cm.
\(A = \tfrac{1}{2} \times 10 \times 5 = 25\).
Answer: \(25\text{ cm}^2\)
Examiner trap
Using the sloping side instead of the perpendicular height: \(\tfrac{1}{2} \times 10 \times 7 = 35\) is wrong. The height is always the one that meets the base at a right angle.
Section 02
Parallelograms and trapezia.
Cut the corner off a parallelogram and slide it across: you get a rectangle with the same base and height, so the area is \(bh\). A trapezium averages its two parallel sides, then multiplies by the distance between them (Fig. 1).
Worked example Parallelogram
A parallelogram has base 8 cm, sloping sides 6 cm and perpendicular height 5 cm. Find its area and perimeter.
Area \(= b \times h = 8 \times 5 = 40\) (the 6 cm is not the height).
Perimeter \(= 8 + 6 + 8 + 6 = 28\) (here the sloping sides ARE needed).
Answer: Area \(40\text{ cm}^2\), perimeter 28 cm
Worked example Trapezium
Find the area of the trapezium in Fig. 1: parallel sides 6 cm and 10 cm, height 4 cm.
Add the parallel sides: \(6 + 10 = 16\).
Halve: \(16 \div 2 = 8\) (the average width).
Multiply by the height: \(8 \times 4 = 32\).
Answer: \(32\text{ cm}^2\)
Your turn
A trapezium has parallel sides 5.5 cm and 9 cm, and the distance between them is 6 cm. Find its area.
In a trapezium the two sides you add must be the PARALLEL ones, and \(h\) is the distance between them. Multiplying all three numbers together, or forgetting the half, are the usual slips.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Which area formula is printed in the 0580 List of formulas?
Quick check+10 XP
A parallelogram has base 9 cm and perpendicular height 4 cm. What is its area?
Quick check+10 XP
A triangle has area \(42\text{ cm}^2\) and base 12 cm. What is its perpendicular height?
Quick check+10 XP
Which length is the \(h\) in the trapezium formula?
Section 03
Working backwards.
Many questions give the area and ask for a missing length. Write the formula, put in what you know, and solve the equation.
Worked example Missing side of a rectangle
A rectangle has area \(84\text{ cm}^2\) and width 6 cm. Find its perimeter.
\(84 = l \times 6\), so \(l = 84 \div 6 = 14\) cm.
Perimeter \(= 2(14 + 6) = 40\).
Answer: 40 cm
Worked example Height of a trapezium
A trapezium has area \(60\text{ cm}^2\) and parallel sides 7 cm and 13 cm. Find the distance between the parallel sides.
\(60 = \tfrac{1}{2}(7 + 13)h = 10h\).
\(h = 60 \div 10 = 6\).
Answer: 6 cm
Worked example Height of a triangle
A triangle has area \(30\text{ cm}^2\) and base 12 cm. Find its perpendicular height.
\(30 = \tfrac{1}{2} \times 12 \times h = 6h\).
\(h = 30 \div 6 = 5\).
Answer: 5 cm
Your turn
A square has perimeter 36 cm. Find its area.
Each side is \(36 \div 4 = 9\) cm, so the area is \(9 \times 9 = 81\text{ cm}^2\).
Examiner trap
Keep units straight: perimeter is a length (cm), area is in square units (cm2). Writing 40 cm2 for a perimeter loses the unit mark.
Stage 2 · Exam
Exam-style questions
Unlocks once the four checks above are done. Worth more XP, written to IGCSE Paper 2 and Paper 4 standard.
Finish the four checks above to unlock the exam questions
Exam style+20 XP
A trapezium has parallel sides of 8 cm and 14 cm. The perpendicular distance between them is 5 cm. Find its area.
Exam style+20 XP
A parallelogram has base 12 cm, sloping sides of 7 cm and perpendicular height 6 cm. Find its area.
Exam style+20 XP
The length of a rectangle is 3 times its width. Its perimeter is 48 cm. Find its area.
Exam style+20 XP
A rectangular lawn is 15 m by 8 m. One box of fertiliser covers \(25\text{ m}^2\). How many boxes must be bought to cover the whole lawn?
Skill unlocked
Areas and perimeters, done.
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