A triangular pyramid paperweight with a base area of 18 square inches and a height of 9 inches holds 18 times 9 divided by 3, which is 54 cubic inches of glass, and that divided-by-three is the whole story of this shape. A triangular pyramid, also called a tetrahedron, is a triangle with an apex above it: one triangular base plus three triangular faces meeting at a single point. Its volume equals one third the base area times the height, the same one-third rule that governs every pyramid and every cone. Three pyramids with the same base and height exactly fill the prism built on that base, which is where the one third comes from, and it is the reason a pyramid always holds far less than the box it appears to fill.

What counts as a triangular pyramid, exactly

A triangular pyramid has 4 triangular faces, 6 edges, and 4 vertices, the simplest possible three-dimensional solid, which is why mathematicians call it a simplex. Any one of its four faces can serve as the base, and the height is then the perpendicular distance from the opposite vertex, the apex, straight down to the base plane. That word perpendicular does real work: the height is the straight drop, not the slanted edge running down a face, and not the slant height measured along a triangular face. A regular tetrahedron is the special case where all four faces are identical equilateral triangles, so every edge has the same length, and a single number then determines the volume, the surface area, and the height. Most real triangular pyramids are not regular: a paperweight, a pyramid-cut gemstone, and a concrete pyramid footing all have a triangular base that is not equilateral, and they are handled by the general formula, not the regular shortcut.

The volume formula, and why one third

The volume of a triangular pyramid equals one third times the base area times the height. In symbols, with B for the triangular base area and h for the perpendicular height, volume equals B times h divided by 3. The one third is not a fudge factor: take the prism with the same triangular base and the same height, and exactly three copies of the pyramid fill it, because the pyramid tapers from full cross-section at the base to a point at the apex, averaging one third of the prism's cross-section along the way. The cone follows the same rule for the same reason, which is why the volume of a cone is also one third times base area times height. Students who remember the pyramid as the prism divided by three rarely forget the division, and forgetting it is the most expensive mistake in this topic, because it triples the answer and triples any material order built on it.

Worked example: the glass paperweight, step by step

Start with the paperweight from the opening paragraph and run every step in the open. The triangular base area is 18 square inches and the vertical height is 9 inches, so their product is 18 times 9, which is 162. Divide by 3 and the volume is 54 cubic inches of glass. On this calculator, type 18 in the Variable A box and 9 in the Variable B box, and the Result box shows 162, the product of base area and height. Divide that Result by 3 yourself and you have the 54 cubic inches of pyramid volume, before any bevels or rounding of the real object are counted. The calculator performs the key multiplication; the division by 3 is a one-step finish you do by hand, and keeping that division visible is what stops the classic triple-order mistake. Try the guardrail check: the prism on the same base and height would hold 162 cubic inches, so a pyramid answer above 162 is impossible, and an answer of 162 means the division was skipped.

Worked example: a trophy base and a Heron base

A trophy maker sizes a triangular pyramid base with a triangular face 16 inches wide and 11 inches tall, perpendicular height of the triangle itself. The base area is one half times 16 times 11, which is 88 square inches. With a pyramid height of 7 inches, the product is 88 times 7, which is 616, and divided by 3 the volume is about 205.33 cubic inches of trophy material. A second example shows the Heron route for a base known only by its sides. A triangular base with sides of 9, 10, and 17 inches has a semi-perimeter of 18 inches, and its area is the square root of 18 times 9 times 8 times 1, the square root of 1,296, which is exactly 36 square inches. With a pyramid height of 13 inches, the product is 36 times 13, which is 468, and divided by 3 the volume is 156 cubic inches. Notice how the two examples use different triangles on purpose: the trophy triangle is a base-and-height triangle, while the Heron triangle is a sides-only triangle, and both feed the same one-third formula once the base area is known.

Pyramid against prism: the same base, one third the volume

The cleanest way to feel the one third is to put the two solids side by side. Take a triangular base area of 42 square feet and a height of 10 feet. The prism built on that base holds 42 times 10, which is 420 cubic feet. The pyramid built on the same base and height holds 420 divided by 3, which is 140 cubic feet. That gap explains why pyramid roofs, pyramid tents, and pyramid packaging look spacious and hold less than expected: the taper removes two thirds of the prism's volume, and most of what it removes sits near the top, exactly where headroom and usable space are already scarce. A buyer comparing a prism-shaped storage box with a pyramid-shaped one of the same footprint and height should therefore expect the pyramid to hold one third as much, not roughly as much, and a concrete order written from the prism number will over-order by a factor of three.

The regular tetrahedron: one edge determines everything

When all four faces are equilateral triangles with edge length a, the pyramid becomes a regular tetrahedron, and its measurements collapse into compact formulas. The perpendicular height is a times the square root of 6 divided by 3, which is about 0.8165 times the edge. The volume is a cubed times the square root of 2 divided by 12, which is about 0.11785 times a cubed. For an edge of 6 inches, the height is about 4.90 inches and the volume is 216 times the square root of 2 divided by 12, which is 18 times the square root of 2, about 25.46 cubic inches. For an edge of 4 inches, the volume is 64 times the square root of 2 divided by 12, about 7.54 cubic inches. For an edge of 10 inches, the volume is 1,000 times the square root of 2 divided by 12, about 117.85 cubic inches. Doubling the edge from 4 inches to 8 inches multiplies the volume by 8, from about 7.54 to about 60.34 cubic inches, because volume scales with the cube of the edge. The surface area of a regular tetrahedron is the square root of 3 times the edge squared: for an edge of 6 inches, that is about 62.35 square inches across four identical faces. Chemists meet this shape in tetrahedral molecules such as methane, where four bonds point to the corners of a tetrahedron at angles of about 109.5 degrees, and jewelers meet it in pyramid-cut stones, where the regular formulas price the rough.

Running the formula backward

The formula rearranges cleanly because it is one multiplication and one division. If a pyramid holds 156 cubic inches and its height is 13 inches, the base area must be 3 times 156 divided by 13, which is 36 square inches, the Heron triangle from earlier, run in reverse. If a pyramid holds 54 cubic inches on a base area of 18 square inches, its height must be 3 times 54 divided by 18, which is 9 inches, the paperweight numbers again. For a regular tetrahedron, the edge comes from the volume by reversing the cube: the edge equals the cube root of 6 times the square root of 2 times the volume. For a volume of about 25.46 cubic inches, 6 times the square root of 2 times 25.46 is 216, and the cube root of 216 is exactly 6 inches, a satisfying closed loop that doubles as an error check. If a backward run returns a base area larger than the triangle you measured, or a height longer than the object's tallest edge, one of the inputs is wrong, and the usual suspect is a slant height entered as the vertical height.

Worked example: a concrete pyramid footing

A pyramid footing for a signpost or a sculpture base shows the formula at job scale. Suppose the triangular base area is 28 square feet and the vertical height is 6 feet. The product is 28 times 6, which is 168, and divided by 3 the volume is 56 cubic feet of concrete. In cubic yards, dividing by 27 gives about 2.07 cubic yards, and at roughly 0.6 cubic feet per 80-pound bag of premixed concrete, the order is 56 divided by 0.6, about 93.3 bags, so 94 bags before waste, with a sensible order adding 10 percent, about 103 bags. The footing's taper is not decoration: a pyramid footing spreads load over a wide base while using one third of the concrete a prism footing of the same footprint and height would need, which is exactly why the shape appears under monuments and signposts. Measure the height vertically from the base plane to the apex, not along the sloped face, or the concrete order inherits the slant's extra length.

How to use this calculator, honestly

This calculator performs the key multiplication at the heart of the triangular pyramid formula. Type the triangular base area in the Variable A box and the pyramid's vertical height in the Variable B box, and the Result box shows their product. For the paperweight, 18 and 9 give 162, and dividing by 3 gives 54 cubic inches. For the trophy, 88 and 7 give 616, and dividing by 3 gives about 205.33 cubic inches. For the Heron base, 36 and 13 give 468, and dividing by 3 gives 156 cubic inches. The tool does not compute the triangular base area for you, and it does not divide by 3, so both of those steps stay visible: find the base area by the base-and-height route or Heron's formula, multiply on the tool, and divide the Result by 3 by hand. Keep the area and the height in matching units, square feet with feet, square inches with inches, and run the backward check before ordering materials.

Five mistakes that cost real money

The first mistake is forgetting to divide by 3, which triples the volume and triples the material order. The second is entering a slant edge or slant height as the vertical height, which overstates the height and the volume, because every slant measurement is longer than the perpendicular drop. The third is using the triangle's perimeter instead of its area, which mixes a length with an area and produces units that mean nothing. The fourth is mixing units, square inches of base area with a height in feet, which corrupts the answer by a factor of 12. The fifth is assuming every triangular pyramid is regular: the one-edge formulas apply only when all four faces are equilateral triangles, and applying them to an ordinary pyramid gives a volume for a different solid than the one being built. Each of these is caught by the same two habits: name every measurement before entering it, and check that the pyramid answer is exactly one third of the prism on the same base and height.

Related calculators on this site

If the triangular pyramid is part of a bigger job, three companion tools carry the next step. The triangular prism calculator handles the untapered cousin, where the same base area and height give three times the pyramid's volume. The volume of a pyramid calculator covers the square-based pyramid, the shape of the great monuments, where the same one-third rule applies to a square base. The triangle area calculator finds the base area from base and height or from three sides, the exact input this page needs, and the volume of a cone page handles the round cousin, where one third times base area times height returns again. Each one uses the same discipline: name the measurement before you enter it, keep units consistent, and check the answer against the prism it came from.

A classroom note on why three pyramids fill a prism

Students sometimes ask whether the one third is a definition or a discovery. It is a discovery you can see: build a triangular prism from clay or cardboard, then cut it, or fill it, with three triangular pyramids that share its base and height, and the three fill it exactly, with no gaps and no overlap. The same demonstration with a square prism and three square pyramids gives the same result, and the cone-and-cylinder version gives it again. That is why every pyramid and cone formula in geometry carries the same one third: the taper, whatever the base shape, averages to one third of the full cross-section over the height, and the prism is the measuring stick against which the taper is judged.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Triangular Pyramid Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.