Why is there a 1/3 in the pyramid formula?

A pyramid is exactly one-third the volume of a prism with the same base and height. This is true for any pyramid shape because it tapers uniformly to a single point. You can verify this by filling a pyramid with sand and pouring it into a prism—you need exactly three pyramid-fulls to fill the prism.

Does the pyramid formula work for any base shape?

Yes. The formula V = \frac{1}{3}Bh works for any pyramid, regardless of whether the base is a triangle, square, pentagon, or any other polygon. You just need to calculate the base area correctly for that shape.

What if I'm given slant height instead of perpendicular height?

Slant height is measured along the face of the pyramid, not vertically. You must find the perpendicular height first using the Pythagorean theorem, treating the perpendicular height as the leg of a right triangle where the slant height is the hypotenuse.

Can a pyramid have a circular base?

A pyramid by definition has a polygon as its base, so a circular base makes it a cone instead. However, the volume formula for a cone is identical: V = \frac{1}{3}Bh, where B is the area of the circle.

Volume of a Pyramid

Find the volume inside a pyramid by multiplying one-third of its base area by its height—the most direct way to measure any pyramid's capacity.

V = \frac{1}{3}Bh

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What each symbol means

What Volume of a Pyramid takes

Symbol	Meaning
$V$	The volume of the pyramid, measured in cubic units (like $\text{cm}^3$ or $\text{m}^3$ )—if you treat $V$ as a linear dimension instead of a volume, your answer will have the wrong units.
$B$	The area of the pyramid's base (the flat polygon at the bottom), measured in square units—using the perimeter instead of area is the most common mistake here.
$h$	The perpendicular height from the base to the apex (point), measured in the same linear units as the base—slant height will give you the wrong answer.

When to use it

Reach for this when you need to find how much space a pyramid encloses.

Level

Usually taught in: Geometry

Worked examples

1. Find the volume of a pyramid with a square base

Problem

A pyramid has a square base with side length 6 cm and height 9 cm. Find its volume.

$B = 6^{2} = 36 \text{ cm}^{2}$
Calculate the base area by squaring the side length of the square base.
$V = \frac{1}{3} \cdot 36 \cdot 9$
Substitute $B = 36$ and $h = 9$ into the pyramid volume formula.
$V = \frac{324}{3} = 108 \text{ cm}^{3}$
Divide $36 \times 9 = 324$ by 3 to get the final volume.

Answer: $V = 108 \text{ cm}^{3}$

A square base makes this a straightforward calculation: find the base area by squaring the side length, then apply the formula directly. This is the simplest type of pyramid and a good starting point.

2. Find volume with a triangular base and radicals

Problem

A pyramid has an equilateral triangular base with side 4 m. The height is 5 m. Find the volume. (The area of an equilateral triangle with side

s

\frac{s^2\sqrt{3}}{4}

$B = \frac{4^{2}\sqrt{3}}{4} = \frac{16\sqrt{3}}{4} = 4\sqrt{3} \text{ m}^{2}$
Use the given formula to find the base area, then simplify the fraction by dividing 16 by 4.
$V = \frac{1}{3} \cdot 4\sqrt{3} \cdot 5$
Substitute $B = 4\sqrt{3}$ and $h = 5$ into the volume formula.
$V = \frac{20\sqrt{3}}{3} \text{ m}^{3}$
Multiply $4\sqrt{3} \times 5 = 20\sqrt{3}$ , then apply the factor $\frac{1}{3}$ to get the final answer.

Answer: $V = \frac{20\sqrt{3}}{3} \text{ m}^{3}$

This example uses an equilateral triangle base and involves radicals, teaching you to simplify $\sqrt{3}$ correctly. It shows that the $\frac{1}{3}$ factor multiplies the entire product, not just the base area.

3. Compare pyramid volumes on a road trip

Problem

You're driving between two ancient sites. At the first stop, you measure a pyramid with base area 144 m² and height 20 m. You drive 150 km to the second site, where another pyramid has base area 100 m² and height 24 m. What is the total volume of both pyramids?

$V_{1} = \frac{1}{3} \cdot 144 \cdot 20 = \frac{2880}{3} = 960 \text{ m}^{3}$
For the first pyramid, multiply $144 \times 20 = 2880$ , then divide by 3.
$V_{2} = \frac{1}{3} \cdot 100 \cdot 24 = \frac{2400}{3} = 800 \text{ m}^{3}$
For the second pyramid, multiply $100 \times 24 = 2400$ , then divide by 3.
$V_{\text{total}} = 960 + 800 = 1760 \text{ m}^{3}$
Add the two volumes together to find the combined volume of both pyramids.

Answer: $V_{\text{total}} = 1760 \text{ m}^{3}$

This problem combines two pyramids with different bases and heights, mirroring the kind of comparison you might make on a real trip. It reinforces that the $\frac{1}{3}$ factor is universal, regardless of base shape or size.

Common mistakes

Where Volume of a Pyramid usually goes wrong

The mistake	Why it is wrong	The fix
Students forget to multiply by $\frac{1}{3}$ , writing $V = Bh$ instead of $V = \frac{1}{3}Bh$ .	The factor of $\frac{1}{3}$ is essential to the pyramid formula; without it, you get the volume of a prism with the same base and height, which is three times too large.	Always include the $\frac{1}{3}$ factor. Remember: a pyramid is exactly one-third the volume of a prism with the same base and height.
Using slant height (the distance along the face) instead of perpendicular height in the formula.	The formula requires the perpendicular distance from the base to the apex; slant height is longer and will overestimate the volume.	If given slant height, use the Pythagorean theorem on the right triangle formed by the slant height, perpendicular height, and the distance from the base center to the edge.
Forgetting to find the base area first, and plugging in the perimeter or a single side length as $B$ .	$B$ must be an area (square units), not a length; using a side length gives you a value thousands of times too small and units that do not make sense.	Always calculate the area of the base shape (triangle, square, rectangle, or any polygon) before substituting into the volume formula.

Tips and when to use something else

The factor of $\frac{1}{3}$ appears because a pyramid tapers to a point; if you imagine stacking three identical pyramids base-to-base, they exactly fill a prism with the same base and height.
Use Volume of a Cylinder for round, tube-like solids—it has the same $\frac{1}{3}$ factor only if they have a conical top, but a cylinder uses the full height for a rectangular volume.
Check your base area first by drawing the base separately; getting the base area wrong ruins the entire calculation.
If the pyramid is oblique (apex not directly above the center), the height is still the perpendicular distance; use a coordinate system or drop a perpendicular line to find it.

Related formulas

Area of a TriangleCalculates the area enclosed by a triangle given its base and perpendicular height; essential for geometry, SAT, and ACT math problems involving triangles.Area of a RectangleMultiply the length and width of a rectangle to find the total area of flat space it covers, measured in square units.Volume of a ConeCalculate the space inside a cone by multiplying one-third of the base area by the height; essential for geometry and SAT problems.Volume of a CylinderCalculates the volume of a cylinder—the three-dimensional space inside—used for any cylindrical container or object in math and engineering.Pythagorean TheoremFind the length of any side of a right triangle given the other two sides, or check whether a triangle is a right triangle.

Frequently asked questions

Why is there a 1/3 in the pyramid formula?: A pyramid is exactly one-third the volume of a prism with the same base and height. This is true for any pyramid shape because it tapers uniformly to a single point. You can verify this by filling a pyramid with sand and pouring it into a prism—you need exactly three pyramid-fulls to fill the prism.
Does the pyramid formula work for any base shape?: Yes. The formula $V = \frac{1}{3}Bh$ works for any pyramid, regardless of whether the base is a triangle, square, pentagon, or any other polygon. You just need to calculate the base area correctly for that shape.
What if I'm given slant height instead of perpendicular height?: Slant height is measured along the face of the pyramid, not vertically. You must find the perpendicular height first using the Pythagorean theorem, treating the perpendicular height as the leg of a right triangle where the slant height is the hypotenuse.
Can a pyramid have a circular base?: A pyramid by definition has a polygon as its base, so a circular base makes it a cone instead. However, the volume formula for a cone is identical: $V = \frac{1}{3}Bh$ , where $B$ is the area of the circle.

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Reviewed 2026-09-18