Pyramid Frustum Calculator: Volume & Base Areas

The pyramid frustum's three measures

A pyramid frustum is a square pyramid with its tip cut off by a plane parallel to the base — a smaller top square of edge $a$ and a larger bottom square of edge $b$, joined by a tapering side of height $h$. Given $a$, $b$, and $h$, the top area, bottom area, and volume all follow:

$A_1 = a^2 \qquad A_2 = b^2$ $V = \frac{h}{3}\left(a^2 + ab + b^2\right)$

This is the simplest solid in its family: no $\pi$ anywhere (a square base, not a circle), and no square root anywhere in these three measures — unlike the plain square pyramid's slant height or the conical frustum's slant height, $A_1$, $A_2$, and $V$ are a plain rational expression in $a$, $b$, and $h$.

Worked example: a = 3, b = 6, h = 4

$A_1 = 3^2 = 9 \qquad A_2 = 6^2 = 36$ $V = \frac{4}{3}\left(9 + 3 \times 6 + 36\right) = \frac{4}{3}(63) = 84$

Because $a = 3$, $b = 6$, and $h = 4$ are all whole numbers, every measure here comes out EXACT — the volume needs no rounding at all.

Worked example: a = 0, b = 6, h = 4 (degenerates to a full pyramid)

$A_1 = 0^2 = 0 \qquad A_2 = 6^2 = 36$ $V = \frac{4}{3}\left(0 + 0 + 36\right) = \frac{4}{3}(36) = 48$

With the top edge shrunk to a point, this is exactly a square pyramid with base edge 6 and height 4 — the volume formula $V = \frac{h}{3}(0 + 0 + b^2) = \frac{1}{3}b^2h$ matches the pyramid's own formula precisely.

Worked example: a = 1, b = 2, h = 1 (volume not a whole number)

$A_1 = 1^2 = 1 \qquad A_2 = 2^2 = 4$ $V = \frac{1}{3}\left(1 + 1 \times 2 + 4\right) = \frac{1}{3}(7) \approx 2.3333$

Even though $a$, $b$, and $h$ are all whole numbers, the volume isn't — $3$ doesn't divide $h(a^2+ab+b^2) = 7$ evenly. The top and bottom areas, by contrast, are always exact whole numbers whenever $a$ and $b$ are, since squaring a whole number always stays whole.

Common mistakes

• Using the area-only shorthand with edge lengths. $V = \frac{h}{3}(a + b + \sqrt{ab})$ is only correct when $a$ and $b$ are already the two base areas — with edge lengths, the correct term is $a^2 + ab + b^2$, not $a + b + \sqrt{ab}$.
• Forgetting the middle term $ab$. The volume is a weighted combination of both areas plus their geometric mean, not just $A_1 + A_2$ — dropping $ab$ understates the true volume.
• Misspelling "frustum" as "frustrum." The correct spelling has no second "r" after the "t."

Where this shows up

• Grain silos, plant pots, lampshades: any container with a square cross-section that tapers from a wide base to a narrower top (or vice versa) is a pyramid frustum — volume tells you capacity.
• Sanity-checking against simpler solids: since $a = 0$ gives a square pyramid and $a = b$ gives a rectangular prism, you can always verify a frustum calculation by checking it against one of those two familiar shapes.

A pyramid frustum is a square pyramid truncated parallel to its base — knowing the top edge, bottom edge, and height is always enough to find the top area, bottom area, and volume, with the square pyramid and the rectangular box sitting at its two limiting cases.