Volume of a Pyramid

\( V=\tfrac{1}{3}\,\text{area of base}\times h \)
Geometry GCSE
Question 10 of 20
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\( Triangular pyramid: base area=15 cm², h=18 cm. Volume? \)

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
\( V=1/3×base area×height. \)

Explanation

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Statement

The volume of a pyramid is given by:

\[ V = \tfrac{1}{3} \times \text{area of base} \times h \]

where the base can be any polygon (square, rectangle, triangle, etc.), and \(h\) is the vertical height from the base to the apex.

Why it’s true

  • A pyramid is like a prism but tapering to a point.
  • Its volume is exactly one third of the volume of a prism with the same base and height.
  • This result comes from calculus or dissection proofs (comparing cubes and pyramids).

Recipe (how to use it)

  1. Find the area of the base (depending on its shape).
  2. Multiply the base area by the perpendicular height.
  3. Divide by 3.
  4. Answer in cubic units.

Spotting it

Pyramids have polygon bases and triangular sides that meet at a single apex.

Common pairings

  • Square-based pyramids are common in GCSE exams.
  • May be asked in comparison with prisms or cones.

Mini examples

  1. Square base: side=6 cm, h=9 cm → base area=36, volume=1/3×36×9=108.
  2. Rectangular base: 8×5=40 cm², h=12 cm → volume=1/3×40×12=160.

Pitfalls

  • Forgetting the 1/3 factor.
  • Using slant height instead of vertical height.
  • Miscomputing the base area if base isn’t square.

Exam strategy

  • Draw a sketch to clarify base and height.
  • Check whether the given height is slant or perpendicular.
  • Keep answers in exact form unless decimals are asked.

Summary

The volume of a pyramid is one third of the base area times the perpendicular height.

Worked examples

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  1. \( Square pyramid: base side 6 cm, h=9 cm. \)
    1. \( Base area=6×6=36 \)
    2. \( V=1/3×36×9=108 \)
    Answer: 108 cm³
  2. \( Rectangular pyramid: base 8×5 cm, h=12 cm. \)
    1. \( Base area=40 \)
    2. \( V=1/3×40×12=160 \)
    Answer: 160 cm³
  3. \( Triangular pyramid: base triangle 6×4 cm, area=12, h=10 cm. \)
    1. \( V=1/3×12×10=40 \)
    Answer: 40 cm³
  4. \( Square pyramid: base side 10 cm, h=15 cm. \)
    1. \( Base=100 \)
    2. \( V=1/3×100×15=500 \)
    Answer: 500 cm³
  5. \( Rectangular pyramid: base 7×9 cm, h=18 cm. \)
    1. \( Base=63 \)
    2. \( V=1/3×63×18=378 \)
    Answer: 378 cm³
  6. \( Square pyramid: base side 4 cm, h=12 cm. \)
    1. \( Base=16 \)
    2. \( V=1/3×16×12=64 \)
    Answer: 64 cm³
  7. \( Triangular pyramid: base 10×8 cm, area=40, h=9 cm. \)
    1. \( V=1/3×40×9=120 \)
    Answer: 120 cm³
  8. \( Square pyramid: base 20 cm, h=30 cm. \)
    1. \( Base=400 \)
    2. \( V=1/3×400×30=4000 \)
    Answer: 4000 cm³
  9. \( Rectangular pyramid: base 12×5 cm, h=20 cm. \)
    1. \( Base=60 \)
    2. \( V=1/3×60×20=400 \)
    Answer: 400 cm³
  10. General pyramid with base area B and height h.
    1. \( V=1/3×B×h \)
    Answer: 1/3 Bh