Step-by-step worked examples and graded practice questions on perimeter and area — rectangles, triangles, parallelograms, trapezia and compound shapes.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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Perimeter is the total distance around the outside of a shape, measured in units of length (cm, m). Area is the amount of space inside a shape, measured in square units (cm², m²). Mixing these up is one of the most common ways to lose easy marks, so always check which one the question is asking for.
Perimeter of a rectangle = 2 × (length + width)
Area of a rectangle = length × width
Area of a triangle = ½ × base × perpendicular height
Worked Example 1
A rectangle has length 7cm and width 3cm. Find its perimeter and area.
Find the area of a triangle with base 8cm and perpendicular height 5cm.
1
Area of a triangle = ½ × base × perpendicular height
2
Area = ½ × 8 × 5 = 20cm²
Answer20cm²
The height of a triangle is always the perpendicular distance from the base to the opposite vertex (dashed) — not a slanted side.
Area of parallelograms and trapezia
Parallelograms and trapezia both use a perpendicular height — the shortest distance between the base and the opposite side, measured at a right angle. This is not the same as the length of the slanted side, which is a common source of errors.
Area of a parallelogram = base × perpendicular height
Area of a trapezium = ½ × (sum of parallel sides) × perpendicular height
Worked Example 3
Find the area of a parallelogram with base 9cm and perpendicular height 5cm.
1
Area = base × perpendicular height
2
Area = 9 × 5 = 45cm²
Answer45cm²
The dashed line shows the perpendicular height — the slanted side of the parallelogram is a different length and is never used in the area formula.
Worked Example 4
Find the area of a trapezium with parallel sides 6cm and 10cm, and perpendicular height 4cm.
1
Area = ½ × (sum of parallel sides) × height
2
Area = ½ × (6 + 10) × 4 = ½ × 16 × 4
3
Area = 32cm²
Answer32cm²
A trapezium has exactly one pair of parallel sides, labelled a and b — the perpendicular height h is measured between them.
Compound (composite) shapes
A compound shape is made from two or more simple shapes joined together. The key technique is to split the shape into rectangles and triangles you already know how to work with, calculate each part separately, then add (or subtract) the results.
Worked Example 5
A garden shed cross-section is made from a rectangle 8m wide and 5m tall, with a triangular roof of height 3m sitting on top. Find the total area of the cross-section.
1
Split the shape into two parts: the rectangular body (B) and the triangular roof (A).
2
Area of rectangle B = 8 × 5 = 40m²
3
Area of triangle A = ½ × 8 × 3 = 12m² (base = 8m, the same width as the rectangle)
4
Total area = 40 + 12 = 52m²
Answer52m²
The dashed line shows where to mentally split the shape — this line isn't a real edge, it's a construction line to help with the calculation.
Practice questions
Work through each question before checking the answers. Where a question describes a compound shape, the diagram shown is required to answer it — the dimensions are given only on the diagram, exactly as they'd appear in a real exam.
Foundation (Grade 3–5)
Q1Find the perimeter of a rectangle with length 9cm and width 4cm.Foundation
Q2Find the area of a rectangle with length 12cm and width 5cm.Foundation
Q3Find the area of a triangle with base 10cm and perpendicular height 6cm.Foundation
Q4Find the perimeter of a square with side length 7cm.Foundation
Q5A rectangle has area 48cm² and length 8cm. Find its width.Foundation
Higher (Grade 5–7)
Q6Find the area of a parallelogram with base 11cm and perpendicular height 6cm.Higher
Q7Find the area of a trapezium with parallel sides 8cm and 12cm, and perpendicular height 5cm.Higher
Q8A rectangular garden measures 15m by 8m. Find its perimeter and area.Higher
Q9Find the area of a right-angled triangle whose two shorter sides are 9cm and 12cm.Higher
Q10The diagram shows a compound shape made from a rectangle and a right-angled triangle. Find the total area of the shape.Higher
Diagram for Question 10 (not to scale). A = rectangle, B = right-angled triangle.
Higher — Hard (Grade 8–9)
Q11A trapezium has area 60cm², parallel sides of (2x)cm and (3x)cm, and perpendicular height 8cm. Find x.Grade 8–9
Q12The diagram shows an L-shaped garden. Find its perimeter.Grade 8–9
Diagram for Question 12 (not to scale). The shaded corner is not part of the garden.
Q13A rectangle measuring 10cm by 6cm has the same area as a triangle with base 8cm. Find the triangle's perpendicular height.Grade 8–9
Q14A rectangle has a perimeter of 36cm. Its length is twice its width. Find the area of the rectangle.Grade 8–9
Q15A rectangle has area (x² + 5x)cm² and length (x + 5)cm. Find an expression for its width.Grade 8–9
Answers
Foundation (Q1–Q5)
Q126cm(2 × (9 + 4))
Q260cm²(12 × 5)
Q330cm²(½ × 10 × 6)
Q428cm(4 × 7)
Q56cm(48 ÷ 8)
Higher (Q6–Q10)
Q666cm²(11 × 6)
Q750cm²(½ × (8+12) × 5)
Q8Perimeter = 46m, Area = 120m²
Q954cm²(½ × 9 × 12)
Q1072cm²(rectangle 60 + triangle 12)
Higher — Hard (Q11–Q15)
Q11x = 3(60 = ½ × 5x × 8)
Q1240m(cutting a rectangular notch from a corner never changes the perimeter — use the overall 12m × 8m bounding rectangle: 2 × (12+8))
Q1315cm(rectangle area 60 = ½ × 8 × h)
Q1472cm²(width 6, length 12, from 6w = 36)
Q15x cm(width = (x²+5x)/(x+5) = x(x+5)/(x+5) = x)
Common mistakes
Common Mistake 1
Using the slanted side instead of the perpendicular height
For parallelograms and trapezia, always use the height that meets the base at a right angle. The slanted side is a different length and is never used in the area formula.
Common Mistake 2
Forgetting the ½ in the triangle area formula
Area of a triangle = ½ × base × height. Missing out the ½ — or forgetting to halve at all — is one of the most common lost marks at Foundation tier.
Common Mistake 3
Missing or double-counting an edge in compound shapes
Always split a compound shape into simple parts first, and check every outer edge is accounted for exactly once — it's easy to miss an internal edge or add a length that doesn't belong on the outer perimeter.
Common Mistake 4
Mixing up perimeter and area
Perimeter is a length (cm, m) and area is a squared measurement (cm², m²). Writing the wrong units — or answering the wrong one entirely — loses marks even when the calculation is correct.
Exam tips
💡 Exam Tip 1
Always split compound shapes before calculating
Draw a dashed line to divide the shape into rectangles and triangles, label each part, then work out each area separately before combining them.
💡 Exam Tip 2
Write the formula before substituting numbers
Starting with "Area = ½ × base × height" before putting numbers in secures method marks even if your final answer has an arithmetic slip.
💡 Exam Tip 3
Re-read the question before you start
Check whether the question wants perimeter (distance around) or area (space inside) — it's easy to calculate the wrong one under time pressure.
💡 Exam Tip 4
For notched rectangles, use the bounding dimensions
Cutting a rectangular corner notch out of a shape doesn't change its perimeter — the length removed from two sides is always replaced by the two new internal edges. Use the overall bounding rectangle's dimensions.
Want to improve your grade faster?
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