Key takeaway: The volume of a pyramid is one third of the base area times the perpendicular height, which equals the volume of a prism with the same base and height divided by three.
Grade 10 – GeometryG-GMD.A.3Geometry and Measurement
Definition of Volume of a Pyramid
The volume of a pyramid measures how much space it fills. A pyramid with any polygon as its base, and a tip, the apex, above it, fills exactly one third of the prism that has the same base and the same height.
That one third can be seen by cutting a cube into three identical pyramids that meet at one corner. The height must be the perpendicular distance from the base to the apex, not the slanted edge or face. The formula works for square, rectangular, triangular and every other polygonal base.
Formula and notation
The volume of a pyramid with base area \(B\) and perpendicular height \(h\) is:
\[ V = \frac{1}{3}\,B\,h \]
For a rectangular base with length \(\ell\) and width \(w\), the base area is \(B = \ell w\), so \(V = \frac{1}{3}\,\ell\, w\, h\).
Key facts
- Three pyramids with the same base and height fill one matching prism.
- The height is perpendicular to the base, drawn from the apex straight down.
- Any polygon can serve as the base; only its area matters.
- Doubling the height doubles the volume.
Volume of a Pyramid example
A pyramid-shaped tent has a rectangular floor 14 ft long and 9 ft wide, and its peak is 11 ft directly above the floor. How much air does it enclose?
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Base area: \(B = 14 \times 9 = 126\) ft².
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Set up: \(V = \frac{1}{3}(126)(11)\).
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Compute: \(126 : 3 = 42\), then \(42 \times 11 = 462\) ft³.
The tent encloses 462 ft³ of air.
Common mistake
Watch out
Gwen multiplies base area by height, \(126 \times 11 = 1{,}386\) ft³, and forgets the factor of one third.
Why it is wrong
The volume of a pyramid is not that product, which is the volume of a box with the same floor and height. A pyramid fills only a third of it, so divide by 3 to get 462 ft³. A quick sense check: the pyramid must hold less than the 1,386 ft³ box around it.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement