Triangular Pyramid Volume Formula & Calculator

This calculator works out the volume of a triangular pyramid (tetrahedron) when you enter the base triangle’s base length, its height, and the pyramid’s overall height. You get the volume in cubic units instantly, using the standard formula V = (1/6) × b × h_base × h_pyramid.

Formula

The volume of a triangular pyramid is one-third of the base area times the pyramid’s height. Since the base is a triangle, its area is (1/2) × base length × base height. Combining these gives:

V = (1/6) × b × h_b × h_p

• b – length of one side of the triangular base (same unit as the other lengths)
• h_b – perpendicular height of the base triangle, measured from side b to the opposite vertex
• h_p – perpendicular height of the pyramid, measured from the base plane straight up to the apex

All three lengths must be in the same unit. The result is in cubic units (e.g., m³, cm³, ft³).

Worked Example

Using the calculator defaults: base length = 6 m, base triangle height = 4 m, pyramid height = 9 m.

1. Base area = (1/2) × 6 × 4 = 12 m²
2. Volume = (1/3) × base area × pyramid height = (1/3) × 12 × 9 = 36 m³

Using the combined formula: V = (1/6) × 6 × 4 × 9 = 36 m³. The calculator shows 36.00 m³.

When to Use This Formula

Use this whenever you need the space inside a triangular pyramid – for example, estimating the volume of a tent with a triangular base, a wedge-shaped container, or a decorative solid. The formula applies to any triangular pyramid, including irregular ones, as long as you know the base triangle’s dimensions and the perpendicular height to the apex.

If your pyramid has a square, rectangular, or other polygonal base, use the corresponding volume formula: V = (1/3) × base area × height.

How to Measure the Heights Correctly

The most common error is mixing up the two heights. The base triangle height lies flat in the base plane – it is the distance from the chosen base side to the opposite vertex of the triangle. The pyramid height is vertical – it goes from the base plane straight up to the apex, not along a slanted edge.

If you only have the slant height (along a triangular face) or the edge length, you must calculate the perpendicular height first using the Pythagorean theorem, then apply the volume formula.

Common Mistakes and Unit Issues

• Mixing units: Convert all lengths to the same unit before multiplying. If base is in meters and height in centimeters, the result will be off by a factor of 10⁶.
• Forgetting the 1/6 factor: The formula is not just base × height; the 1/6 is essential because of the triangular base and the pyramid’s taper.
• Using slant height instead of perpendicular height: This overestimates the volume. Always use the vertical height from the base plane to the apex.
• Rounding too early: Keep full precision in intermediate steps, then round the final answer to a sensible number of decimal places.

Limitations and When It Doesn’t Apply

This formula assumes a straight pyramid – the apex is directly above any point inside the base triangle, so the height is perpendicular. If the apex leans outside the base (an oblique pyramid), the perpendicular height is still measured from the base plane to the apex, and the same formula works. However, if you do not have the perpendicular height, you cannot use this simple formula; you would need more advanced methods (e.g., cross products with coordinates).

Also, the base must be a flat triangle. If the base is curved or the sides are not planar, this is not a triangular pyramid.

Frequently Asked Questions

What is the difference between a triangular pyramid and a tetrahedron?

A tetrahedron is a triangular pyramid where all four faces are equilateral triangles. The volume formula is the same, but for a regular tetrahedron you can also use V = a³ / (6√2), where a is the edge length. The calculator here works for any triangular base.

Can I use this formula if the base triangle is not equilateral?

Yes. The formula works for any triangle as long as you know the base length and the corresponding height of that triangle. You do not need equal sides.

How do I find the volume if I only know the base area and the pyramid height?

Then use V = (1/3) × base area × pyramid height. That is the simpler form. The calculator’s formula just expands base area into (1/2) × b × h_b.

What if my pyramid has a square base?

Use V = (1/3) × side² × height. For a rectangular base, V = (1/3) × length × width × height. This calculator is specifically for triangular bases.

Why is there a 1/6 in the formula?

Because the base area is half of a rectangle (1/2), and a pyramid is one-third of a prism with the same base (1/3). Multiplying 1/2 × 1/3 gives 1/6.

Can I use this for a pyramid with a slanted apex?

Yes, as long as you measure the perpendicular height from the base plane to the apex. The volume of any pyramid (straight or oblique) is (1/3) × base area × perpendicular height.

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in b | Base length | m | 6 | 0
in hb | Base triangle height | m | 4 | 0
in hp | Pyramid height | m | 9 | 0
out vol | Volume | m³ | (1/6) * b * hb * hp | 2