Pyramid volume and triangular faces
Distinguish perpendicular solid height from sloping face height.
Core and Extended.
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.