Volume of a Pyramid
In short: The volume of a pyramid is a third of its base area multiplied by its straight-up height, which makes it a third of the box standing on the same base.
GeometryGeometryG-GMD.A.3
What is Volume of a Pyramid?
The volume of a pyramid follows \(V = \frac{1}{3}Bh\), where \(B\) is the area of the floor and \(h\) is the straight-up distance from the floor to the apex. The floor can be a square, rectangle, triangle or any polygon.
The fraction comes from a classic fact: three identical pyramids fit together to fill one prism with the same base and height. Cavalieri’s principle stretches the formula to pyramids whose apex leans off-center, as long as the height is measured vertically and not along a slanted face.
How to spot it
- A solid has a flat polygon at the bottom and triangular faces meeting at a single point at the top.
- A tent, roof, monument or paperweight with that shape needs its capacity or filling amount.
- A problem lists a base length, base width and a vertical height, and the volume of a pyramid is the goal.
- The base area and the volume are given and a missing height is requested.
The method
- Compute the base area \(B\) from the polygon’s own formula, such as length times width.
- Find the perpendicular height \(h\), not the slanted edge or the face height.
- Multiply to get the volume of a pyramid: \(V = \frac{1}{3} \times B \times h\).
- State the answer in cubic units, and solve \(h = \frac{3V}{B}\) if the height is the unknown.
Volume of a Pyramid: solved example
Problem
A pyramid-shaped camping tent has a rectangular floor of 10 feet by 7 feet and a center pole 12 feet tall. How many cubic feet of air does it hold?
Solution
The base area is \(B = 10 \times 7 = 70\) square feet, and the pole gives the perpendicular height \(h = 12\).
\(V = \frac{1}{3} \times 70 \times 12 = \frac{840}{3} = 280\). The tent holds 280 cubic feet of air, a third of the 840 cubic feet of a box with the same base and height.
Try it
Question 1
A glass paperweight is a square pyramid with base sides of 12 cm and a vertical height of 5 cm. Find its volume.
Show the answer
\(B = 12^2 = 144\), so \(V = \frac{1}{3}(144)(5) = 240\) cubic cm.
Question 2
A pyramid has volume 252 cubic inches and a base area of 63 square inches. How tall is it?
Show the answer
\(h = \frac{3V}{B} = \frac{756}{63} = 12\) inches.
Volume of a Pyramid vs. Rectangular Prism
A prism has two matching bases and its volume is the full \(Bh\). A pyramid narrows to one point, and its volume is only a third of that. A 10 by 7 box of height 12 holds 840 cubic units; the pyramid on the same base with the same height holds 280.
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