This pyramid volume calculator works out the volume of a right pyramid on five different bases: square, rectangular, equilateral triangle, regular pentagon and regular hexagon. Alongside the volume it returns the base area, the slant height of a face, the lateral surface area of the sloping sides and the total surface area including the base. The substitution line shows the arithmetic with your own numbers, so a square-based pyramid with a 6 cm side and a 10 cm height reads V = ⅓ × 36 × 10 = 120 cm³.
Arb Digital built the multi-base version because most pyramid pages only handle a square and then stop. Roof structures, hoppers, gemstone cuts, tent forms and packaging designs use triangular and hexagonal bases as often as square ones, and the underlying formula does not care which polygon sits at the bottom.
What This Pyramid Volume Calculator Does
The core calculation is the same for every base: volume equals one third of the base area times the perpendicular height. What changes between shapes is how the base area is found. For a rectangle it is length times width. For a regular polygon with n sides of length a, the area is half the perimeter times the apothem, where the apothem is the distance from the centre of the polygon to the midpoint of a side.
Surface area is where the shapes genuinely diverge. On a regular base, all the triangular faces are identical, so the lateral area is half the base perimeter times the slant height. On a rectangular base there are two different slant heights, one for the pair of faces that meet the long sides and another for the pair that meet the short sides, so the calculator computes both and adds the four faces separately. Getting that right is the difference between a usable answer and a rough guess.
Volume is also reported as capacity in litres and US gallons, because pyramid and hopper shapes are common in storage and material handling, and nobody orders material by the cubic centimetre.
How to Use It
- Choose the base shape. The input labels change to match: a single side length for regular polygons, a length and a width for a rectangle.
- Enter the base dimensions in a single consistent unit.
- Enter the perpendicular height. This is the vertical distance from the base plane to the apex, not the sloping distance up a face.
- Pick your unit. Areas come back squared and volume cubed, with capacity underneath.
- Read the working line to confirm which base area and which slant height the calculation used.
The Formula: How a Pyramid Is Calculated
The universal volume formula is V = ⅓ × B × h, where B is the area of the base and h is the perpendicular height. It holds for any base at all, regular or irregular, and even for oblique pyramids whose apex is offset, provided h is still measured perpendicular to the base plane.
- Square base, side a: B = a², so V = ⅓a²h. With a = 6 cm and h = 10 cm, V = ⅓ × 36 × 10 = 120 cm³.
- Rectangular base: B = l × w, so V = ⅓lwh.
- Regular n-gon base, side a: the apothem is a ÷ (2 tan(π/n)) and B = ½ × na × apothem. For a hexagon that reduces to the familiar 2.598a².
- Slant height on a regular base: s = √(h² + apothem²). For the 6 cm square with h = 10 cm, the apothem is 3 cm, so s = √109 = 10.44 cm.
- Lateral area on a regular base: ½ × perimeter × s. Here ½ × 24 × 10.44 = 125.3 cm², and total area is 125.3 + 36 = 161.3 cm².
The rectangular case needs two slant heights: s₁ = √(h² + (w/2)²) for the faces along the length, and s₂ = √(h² + (l/2)²) for the faces along the width. Lateral area is then l × s₁ + w × s₂. Reference definitions for these relations are collected at Wolfram MathWorld's pyramid entry.
Why One Third, and Why It Never Changes
Three identical pyramids can be assembled to fill a cube exactly. That is the physical demonstration behind the one-third factor, and it explains why the same factor appears in the cone formula: any solid that tapers uniformly from a base to a single point holds one third of the prism or cylinder built on the same base with the same height.
This gives a fast check that needs no calculator. Work out the volume of the box or prism that would enclose the pyramid, then divide by three. A pyramid with a 6 by 6 cm base and a 10 cm height sits inside a 360 cm³ prism, so its volume must be 120 cm³. If your answer is not a third of the enclosing prism, the arithmetic is wrong. Multiplying length by width by height gives that enclosing figure in one step.
Because the factor is exact and universal, a pyramid with an irregular base can still be solved: measure the base area however you like, then multiply by height and divide by three. For triangular bases measured by base and perpendicular height rather than side length, the triangle area calculator handles the first step.
Slant Height, Lateral Edge and the Apothem
A pyramid has three different sloping measurements and confusing them produces answers that are wrong by a wide margin. The slant height runs from the apex down the middle of a triangular face to the midpoint of a base edge. The lateral edge runs from the apex to a base corner and is always longer. The apothem is a base measurement, from the centre of the base to the midpoint of a side.
For the 6 cm square pyramid with a 10 cm height, the slant height is 10.44 cm while the lateral edge is √(10² + (3√2)²) = √118 = 10.86 cm. Use the slant height for surface area, because it is the height of each triangular face. Use the lateral edge when you are cutting the frame members that run to the corners, because that is the piece you actually have to fabricate.
Both come from the same right-triangle logic: the apothem pairs with the height to give the slant, and the circumradius pairs with the height to give the lateral edge. If you want to see that step in isolation, the Pythagorean theorem calculator is the underlying tool.
Partial Fill: Why a Hopper Empties Unevenly
Point-down pyramidal hoppers behave the same way conical ones do. Volume scales with the cube of the linear scale factor, so a pyramid filled to half its height from the point contains only one eighth of its full capacity. Filled to three quarters of its height, it holds about 42%. A level gauge marked evenly up the side of such a vessel will consistently mislead anyone reading it as a proportion of contents.
The point-up case is the mirror image and comes up in bulk material piles. A conical or pyramidal stockpile that has lost the top quarter of its height has lost only about 1.6% of its volume, because the missing tip is tiny relative to the wide base. This is why estimating stockpiles from height alone is unreliable and why the base footprint dominates the answer.
To convert a computed volume into a different capacity unit without redoing any geometry, use the volume converter. If the pyramid is a concrete pour such as a footing or a plinth, the concrete calculator turns the volume into bags and cubic yards.
Right Pyramids, Oblique Pyramids and Frustums
This calculator assumes a right pyramid: the apex sits directly above the centre of the base. For an oblique pyramid, the volume formula is unchanged as long as the height is the perpendicular distance from the base plane to the apex, but the surface area figures do not apply, because the triangular faces are no longer congruent and each needs its own calculation.
A frustum is a pyramid with the top cut off parallel to the base, and it is a far more common real shape than a complete pyramid. Truncated hoppers, plinths, lampshades, roof sections and most tapered planters are frustums. Their volume is the full pyramid minus the removed tip, which requires the dimensions of both the top and the bottom face. Do not apply a full-pyramid formula to a truncated shape, or you will overstate the volume, sometimes badly. All unit conversions used here follow NIST's guide to the SI.
Arb Digital designs, writes and ships free calculator pages built to rank, complete with schema markup and the technical SEO underneath.
Browse All Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Using the slant height as the perpendicular height. Volume needs the vertical height. The slant is always longer, so the volume comes out too big.
- Forgetting the one-third factor. Base area times height is the prism. The pyramid is a third of it.
- Applying the single-slant lateral formula to a rectangular base. A rectangular pyramid has two distinct slant heights and four faces in two matching pairs.
- Including the base when it is not there. An open pyramidal hopper or a tent has no base face, so use lateral area only.
- Treating a frustum as a full pyramid. A truncated shape needs both the top and bottom dimensions or the volume is overstated.
Related Free Tools From Arb Digital
For round tapering shapes the cylinder volume calculator and sphere volume calculator cover the curved solids, and the triangle area calculator handles a triangular base measured by base and height. Browse the full free online tools hub for the rest.
Frequently Asked Questions
Volume is one third of the base area times the perpendicular height, V = ⅓Bh. A square pyramid with a 6 cm side and a 10 cm height holds ⅓ × 36 × 10 = 120 cubic centimetres.
Yes. The one-third factor applies to every pyramid regardless of base, and even to oblique pyramids, as long as the height is measured perpendicular from the base plane to the apex.
The slant height runs from the apex down the centre of a triangular face to the midpoint of a base edge. On a regular base it equals the square root of the height squared plus the apothem squared.
The lateral edge runs from the apex to a base corner and is always longer than the slant height. Use slant height for surface area and lateral edge when cutting frame members that meet at the corners.
On a regular base, lateral area is half the base perimeter times the slant height, and total area adds the base. A rectangular base has two different slant heights, so its four faces are calculated in two pairs.
Filled to half its height from the point, a pyramid contains one eighth of its total volume, because volume scales with the cube of the linear fill ratio. Evenly spaced depth markings are misleading on such a vessel.
No. A frustum needs the dimensions of both the top and bottom faces. Treating one as a full pyramid overstates the volume, since the removed tip is not accounted for.
Results are mathematical values rounded for display. Real structures have material thickness, joints and waste allowances, so add a margin before cutting or ordering to these figures.
