GCSE Maths • Foundation & Higher
GCSE Area and Volume Revision Guide
Build confidence with the area of triangles and trapeziums, compound shapes and the volume of prisms. Learn the key formulae, identify perpendicular heights and cross-sections, solve missing-value questions and use square and cubic units correctly.
Clear explanations
Exam practice
Worked examples
What You'll Learn
Calculate Triangle Areas
Use Area = 1/2 × base × perpendicular height and rearrange the formula to find missing bases or heights.
Calculate Trapezium Areas
Identify the two parallel sides and perpendicular height, then use Area = 1/2 × (a + b) × h.
Solve Compound Shapes
Split a compound shape into simpler parts, calculate each area and add or subtract the parts as required.
Find the Volume of Prisms
Calculate the cross-sectional area and multiply it by the prism length.
Use the Correct Units
Write area answers in square units such as cm² and volume answers in cubic units such as cm³.
Key Area and Volume Rules
Triangle area
Area = 1/2 × base × height. The height must be perpendicular to the base.
Trapezium area
Area = 1/2 × (a + b) × h, where a and b are the parallel sides and h is the perpendicular height.
Compound shapes
Split the shape into simple shapes, calculate each area and add the areas. Subtract areas when there is a hole or missing section.
Prism volume
Volume = cross-sectional area × length. Calculate the cross-sectional area before finding the volume.
Units
Area is measured in square units, while volume is measured in cubic units. Keep units consistent throughout a calculation.
Area and Volume Lessons
Area of Triangles
Identify a base and its perpendicular height, apply the triangle area formula and solve missing-value questions.
Area of Trapeziums
Identify the parallel sides and perpendicular height, use the trapezium formula and rearrange it when a length is missing.
Adding Areas in Compound Shapes
Split shapes into rectangles, triangles, trapeziums or semicircles and add the individual areas.
Subtracting Missing Sections
Calculate the area of a whole shape and subtract holes or removed sections.
Volume of Prisms
Find the constant cross-sectional area, multiply by the prism length and use cubic units.
Worked Examples
Example 1: Area of a Triangle
A triangle has a base of 8 cm and a perpendicular height of 5 cm.
1. Write the formula:
Area = 1/2 × base × height.
2. Substitute the values:
Area = 1/2 × 8 × 5.
3. Calculate: Area = 20 cm².
Area = 1/2 × base × height.
2. Substitute the values:
Area = 1/2 × 8 × 5.
3. Calculate: Area = 20 cm².
Example 2: Area of a Trapezium
The parallel sides are 8 cm and 12 cm, and the perpendicular height is 5 cm.
1. Write the formula:
Area = 1/2 × (a + b) × h.
2. Substitute the values:
Area = 1/2 × (8 + 12) × 5.
3. Calculate: Area = 50 cm².
Area = 1/2 × (a + b) × h.
2. Substitute the values:
Area = 1/2 × (8 + 12) × 5.
3. Calculate: Area = 50 cm².
Example 3: Area of a Compound Shape
A 10 cm × 6 cm rectangle is joined to a triangle with base 8 cm and height 5 cm.
1. Rectangle area:
10 × 6 = 60 cm²
2. Triangle area:
1/2 × 8 × 5 = 20 cm²
3. Total area: 60 + 20 = 80 cm².
10 × 6 = 60 cm²
2. Triangle area:
1/2 × 8 × 5 = 20 cm²
3. Total area: 60 + 20 = 80 cm².
Example 4: Volume of a Triangular Prism
A triangular prism has a cross-section with base 6 cm and height 4 cm. Its length is 10 cm.
1. Find the triangular cross-sectional area:
1/2 × 6 × 4 = 12 cm².
2. Use Volume:
cross-sectional area × length.
3. Volume12 × 10 = 120 cm³.
1/2 × 6 × 4 = 12 cm².
2. Use Volume:
cross-sectional area × length.
3. Volume12 × 10 = 120 cm³.
- Common GCSE Mistakes
- Using a sloping side instead of the perpendicular height.
- Forgetting to divide by 2 when calculating the area of a triangle or trapezium.
- Using a non-parallel side in the trapezium formula.
- Splitting a compound shape incorrectly or forgetting to subtract a missing section.
- Using perimeter instead of area when finding the volume of a prism.
- Mixing units or omitting square and cubic units.
- Making arithmetic errors or rearranging a formula incorrectly.
- Exam Tips
- Write the relevant formula before substituting any values.
- Mark the perpendicular height clearly and check that it is at 90° to the base or parallel sides.
- Draw dividing lines on compound-shape diagrams to show the simple shapes.
- Decide whether the component areas should be added or subtracted.
- For a prism, identify the cross-section and calculate its area before multiplying by the length.
- Check every answer for the correct square or cubic units.
- Practise missing-length and missing-area questions using GCSE exam-style and past-paper questions.
Ready to Practice?
Printable Worksheet
Practise triangle and trapezium areas, compound shapes and prism volumes, including missing-value questions and answers.
Interactive Quiz
Test formula recall, perpendicular heights, cross-sections and the correct use of square and cubic units.
Exam-Style Questions
Build confidence with multi-step area, compound-shape and volume problems.
Watch the Area and Volume Tutorial
Follow a clear, step-by-step explanation of equivalent fractions, simplifying, comparing, the four operations, mixed numbers and fractions of amounts.
- Equivalent fractions and simplifying
- Adding, subtracting, multiplying and dividing
- Mixed numbers, improper fractions and fractions of amounts
Continue Your GCSE Maths Revision
Frequently Asked Questions
Area measures the space inside a two-dimensional shape and uses square units. Volume measures the space inside a three-dimensional object and uses cubic units.
The height is the perpendicular distance from the chosen base to the opposite vertex. It must be at a right angle to the base, so a sloping side cannot be used unless it is perpendicular.
Use the two parallel sides, usually labelled a and b, and the perpendicular height between them. Then apply Area = 1/2 × (a + b) × h.
Split the compound shape into simple shapes, calculate each area separately and add the areas. If there is a hole or removed section, subtract its area instead.
Identify the constant cross-section, calculate its area and multiply that area by the prism length. Write the answer in cubic units.
Source Practice Questions and Answers
Area of Triangles
- 1. Find the area of a triangle with base 10 cm and height 8 cm.
- 2. Find the area of a triangle with base 7 cm and height 12 cm.
- 3. Find the area of a triangle with base 16 cm and height 5 cm.
- 4. A triangle has area 36 cm² and base 9 cm. Find the height.
- 5. A triangle has area 50 cm² and height 10 cm. Find the base.
- 6. Find the area of a triangle with base 13 cm and height 6 cm.
Answers: 1. 40 cm²; 2. 42 cm²; 3. 40 cm²; 4. 8 cm; 5. 10 cm; 6. 39 cm²
Area of Trapeziums
- 1. Parallel sides: 6 cm and 14 cm. Height: 5 cm.
- 2. Parallel sides: 10 cm and 18 cm. Height: 4 cm.
- 3. Parallel sides: 12 cm and 20 cm. Height: 7 cm.
- 4. Area = 60 cm². Parallel sides = 6 cm and 9 cm. Find the height.
- 5. Area = 96 cm². Height = 8 cm. One parallel side = 10 cm. Find the other.
- 6. Parallel sides: 9 cm and 15 cm. Height: 6 cm.
Answers: 1. 50 cm²; 2. 56 cm²; 3. 112 cm²; 4. 8 cm; 5. 14 cm; 6. 72 cm²
Compound Shapes
- 1. Two rectangles have areas 20 cm² and 15 cm².
- 2. A rectangle has area 40 cm² and an attached triangle has area 12 cm².
- 3. A rectangle has area 60 cm² and an attached semicircle has area 14 cm².
- 4. A large rectangle has area 100 cm² and a missing rectangle has area 20 cm².
- 5. A trapezium has area 36 cm² and an attached rectangle has area 24 cm².
- 6. Two triangles of area 10 cm² each are attached to a rectangle of area 30 cm².
Answers: 1. 35 cm²; 2. 52 cm²; 3. 74 cm²; 4. 80 cm²; 5. 60 cm²; 6. 50 cm²
Volume of Prisms
- 1. Cross-sectional area = 10 cm², length = 8 cm.
- 2. Cross-sectional area = 16 cm², length = 12 cm.
- 3. Cross-sectional area = 25 cm², length = 6 cm.
- 4. Volume = 240 cm³, length = 12 cm. Find the cross-sectional area.
- 5. Volume = 300 cm³, cross-sectional area = 20 cm². Find the length.
- 6. Cross-sectional area = 14 cm², length = 15 cm.
