Pyramid volume and triangular faces
Distinguish perpendicular solid height from sloping face height.
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Calculate triangle and square areas.
- Identify perpendicular heights.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which area does a square of side 6 cm have?
What you will learn
- Find pyramid volume from base area and perpendicular height.
- Find a regular square pyramid's surface area using face heights.
One third of a matching prism
A pyramid's volume is Ah/3, where A is the base area and h is the perpendicular distance from the apex to the base plane. A square base side 6 cm has area 36 cm²; with height 4 cm the volume is 48 cm³. The factor one third distinguishes a pyramid from a prism with the same base and height.
The perpendicular solid height is often inside the pyramid. A sloping edge from apex to a base vertex, or a face height down to the midpoint of an edge, is a different measurement. If only volume and base area are supplied, recover h = 3V/A. Show which height your calculation requires before substituting.
Count the base and every triangular face
A right regular square pyramid has four congruent triangular faces. If base side b and each face's perpendicular height l are given, total area is b² + 4(bl/2) = b² + 2bl. For b = 6 and l = 5 cm, total area is 36 + 60 = 96 cm².
That face height 5 cm can coexist with solid height 4 cm because half the base side is 3 cm and 3² + 4² = 5². You need this extra right-triangle relationship only when a face height is not supplied. An uncovered base must be excluded if the question asks for roof area or just the lateral triangular faces.
Pause and explain
A pyramid has base area 30 cm² and perpendicular height 9 cm. Find volume.
Worked example
A right regular square pyramid has base side 6 cm, perpendicular height 4 cm and face height 5 cm. Find volume and total area.
Show the worked solution
- Base area is 36 cm², so volume is 36 × 4/3 = 48 cm³.
- Each triangular face area is 6 × 5/2 = 15 cm²; four total 60 cm².
- Add the base: total area is 96 cm².
Answer 48 cm³; 96 cm².
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the volume of a pyramid with base area 24 cm² and height 5 cm.
Give me a hint
Multiply the base area by perpendicular height, then divide by three.
Compare my reasoning
- The matching prism volume is 24 × 5 = 120 cm³.
- The pyramid occupies one third, giving 40 cm³.
- Check that the height is perpendicular to the base plane.
40 cm³.
Look for these in your work
- Used the factor one third.
- Selected the solid's perpendicular height.
A pyramid has volume 72 cm³ and base area 18 cm². Find its perpendicular height.
Give me a hint
Rearrange h = 3V/A.
Compare my reasoning
- Multiply volume by three: 216 cm³.
- Divide by base area 18 cm² to obtain 12 cm.
- Check 18 × 12/3 = 72 cm³.
12 cm.
Look for these in your work
- Rearranged without losing the factor three.
- Verified the height.
A square pyramid roof has base side 8 m and each triangular face height 5 m. Find the roof covering area if the square base is uncovered.
Give me a hint
Only the four triangular faces need covering.
Compare my reasoning
- Each face has area 8 × 5/2 = 20 m².
- Four faces give 80 m².
- Do not add the square base area because the base is not covered.
80 m².
Look for these in your work
- Used the face's perpendicular height.
- Excluded the uncovered base.
Common mistakes
- Using slant height for volume.
- Omitting the volume factor one third.
- Adding a base that is not part of the required covering.
What can you explain now?
Can a pyramid's perpendicular height be replaced by a sloping edge in its volume formula?
Compare with the explanation
No.
Volume uses distance perpendicular to the base plane; a sloping edge generally has a different length.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, label solid height, face height and edge on a pyramid; state which measure belongs in each calculation.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.