Triangular Pyramid Volume Calculator

V = (1/3) * baseArea * height, where the height is the perpendicular distance from the apex to the base plane. The same 1/3 factor and same formula apply to every pyramid (triangular, square, pentagonal) and to every cone (which is a pyramid with a circular base in the limit). The base shape determines how you compute baseArea (Heron's formula for a triangle from three sides, 0.5 * b * h for a triangle from base and height, side^2 for a square, pi * r^2 for a circle), but everything after baseArea is identical. That is what makes the pyramid family worth memorizing as a single rule rather than as a stack of shape-specific formulas.

Yes. A regular tetrahedron has all four faces as congruent equilateral triangles of edge length a, and its volume is V = a^3 / (6 * sqrt(2)), which is roughly a^3 / 8.485. The apex sits a height of a * sqrt(2/3) (equivalently a * sqrt(6) / 3) above the centroid of the base, and the total surface area is sqrt(3) * a^2 (four equilateral faces of area sqrt(3)/4 * a^2 each). If you only have a regular tetrahedron's edge length, the calculator skips the general (1/3) * baseArea * height path and applies these closed forms directly.

Because not every set of three positive numbers can form a real triangle. The triangle inequality requires every side to be strictly less than the sum of the other two. Sides like 1, 2, 5 fail (1 + 2 = 3, well under 5), so no triangle exists and the base area would come out as the square root of a negative number under the radical. Even the borderline case 1, 2, 3 (where 1 + 2 = 3 exactly) is degenerate, producing a collapsed line with zero area and therefore zero pyramid volume. The calculator filters both cases out and returns no result rather than displaying an imaginary or zero volume.

Pyramid height is the perpendicular drop from the apex straight down to the base plane. Slant height is the sloped distance from the apex along a lateral face to the midpoint of a base edge. They form the two legs of a right triangle whose third leg is the perpendicular distance from the base's center (specifically the incenter, for a right pyramid) to that edge. Volume only uses the pyramid height, since (1/3) * baseArea * height is a vertical-stack integral. Lateral surface area uses the slant height, because each triangular face's area is 0.5 * base edge * slant. Confusing the two underestimates surface area whenever the pyramid is shallow (slant much longer than the vertical drop).

From integrating the cross-sectional area along the apex-to-base axis. A pyramid is self-similar in horizontal slices: if you slice it parallel to the base at height z above the base, the slice is a scaled copy of the base with linear scale (h - z) / h, so its area is baseArea * ((h - z) / h)^2. Integrating that from 0 to h gives baseArea * h / 3. The 1/3 is exactly the integral of x^2 from 0 to 1, which equals 1/3. The result depends only on the self-similar scaling, not on the shape of the base, which is why the same 1/3 also appears in the cone volume formula and works for any pyramidal solid (square, hexagonal, oblique, irregular).