Surface Area and Volume • Topic 7 of 7

Pyramids

A pyramid has a polygon base and triangular faces meeting at an apex. Its volume is one-third of the base area times the height: ⅓ × base area × height — the same ⅓ factor as a cone. For a square base of side 6 (area 36) and height 5, the volume is ⅓ × 36 × 5 = 60. Find the base area first, multiply by the height, then take a third. Forgetting the ⅓ is the most common error and triples the answer. The height is the perpendicular distance from base to apex, not a slant edge. The SAT keeps pyramids to square bases with clean numbers.

A square-based pyramid with base side and heightPyramidhbVolume = ⅓ × base area × h

✅ Solved examples

1. A pyramid has a square base of side 6 and height 5. Find its volume.
⅓ × 36 × 5 = 60.
2. A pyramid has a square base of side 3 and height 9. Find its volume.
⅓ × 9 × 9 = 27.
3. A pyramid has a square base of side 4 and height 6. Find its volume.
⅓ × 16 × 6 = 32.
4. A pyramid has a square base of side 5 and height 12. Find its volume.
⅓ × 25 × 12 = 100.

✏️ Practice — try these, take hints as needed

1. A pyramid has a square base of side 3 and height 4. Find its volume.
⅓ × base area × height.
⅓ × 9 × 4.
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12.
2. A pyramid has a square base of side 6 and height 4. Find its volume.
⅓ × 36 × 4.
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—
48.
3. A pyramid has a square base of side 2 and height 9. Find its volume.
⅓ × 4 × 9.
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—
12.
4. A pyramid has a square base of side 9 and height 3. Find its volume.
⅓ × 81 × 3.
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—
81.
5. A pyramid has a square base of side 10 and height 6. Find its volume.
⅓ × 100 × 6.
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—
200.

📝 Topic test — 8 questions

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