Pyramid Volume and Surface Area Calculator
The volume of a pyramid is V = ⅓ × base area × height, and this pyramid calculator works it out the moment you enter the base and the height. Pick a square, rectangular, or triangular base, and you'll get the volume in cubic inches or feet, liters, and gallons, plus the surface area and slant height for square and rectangular pyramids. A pyramid holds a third of the matching prism, and the calculator shows the formula and steps with every result.
- Square, rectangular, triangular
- Volume = ⅓ base × height
- Surface area & slant
- Base area shown
- 14 output units
Last updated June 15, 2026 Method: V = ⅓ × base area × h Reviewed by the Calcowa math team
Enter positive base measurements and a height to see the volume.
Show all volume units
V = ⅓ × 6² × 4 = 48 in³
What is the volume of a pyramid?
The volume of a pyramid is the space it fills, found with V = ⅓ × base area × height, where the height is measured straight up from the base to the apex. For a square base of 6 and a height of 4, the pyramid volume is 48 cubic units.
The one-third is the key. A pyramid fills exactly a third of the prism with the same base and height, so once you've got the base area, you multiply by the height and take a third. Whether the base is a square, a rectangle, or a triangle, you'll use the same rule.
How do you find the volume of a pyramid?
To find the volume of a pyramid, work out the base area, multiply by the height, then take a third. Here's the full sequence:
- 1
Find the base areaWork out the area of the base: side² for a square, length × width for a rectangle, or ½ × base × height for a triangle.
- 2
Multiply by the heightMultiply the base area by the pyramid height, measured straight up to the apex.
- 3
Take one thirdMultiply that by one-third to get the volume.
- 4
Convert the unitsConvert to liters, gallons, or cubic feet if that's what you need.
Volume of a pyramid with a square base
A square-based pyramid, like the ones in Egypt, has a base area of side squared, so its volume is V = ⅓ × a² × h. For a base side of 6 and a height of 4, that's ⅓ × 36 × 4 = 48 cubic units. Set the base to Square, enter the side and the height, and the calculator handles the rest. It's the most common pyramid you'll meet in class.
Rectangular and triangular base pyramids
The same one-third rule covers every base. For a rectangular base, the base area is length × width, so V = ⅓ × l × w × h. For a triangular base, also called a triangular pyramid, the base area is ½ × base × height of the triangle, so V = ⅓ × (½ b × htri) × h. Pick the base shape at the top of the calculator and the inputs change to match.
Why is a pyramid one third of a prism?
It comes down to a neat fact: three identical pyramids fit exactly inside a prism that shares their base and height. Pour one pyramid of water into the matching prism and it fills a third; three fills it to the top. That's why the formula is ⅓ × base area × height rather than the prism's base area × height. Calculus gives the same result by adding up thin slices, but the three-pyramids picture is the one that sticks.
Surface area and slant height of a pyramid
The surface area of a pyramid adds the base to its triangular faces: SA = base area + lateral area. For a square pyramid, the lateral area is 2 × a × the slant height, so SA = a² + 2a l. The slant height l is the distance from the apex down the middle of a face, found from the height and half the base with l = √((a÷2)² + h²). The calculator shows the surface area and slant height for square and rectangular bases, so you'll see them beside the volume.
A pyramid volume example
Say you've got a square-based pyramid with a 9 m base and a height of 12 m. The base area is 9 × 9 = 81 square meters, times the height of 12 gives 972, and a third of that is the volume.
V = ⅓ × 9² × 12 = 324 m³
base area 81 m² × height 12 m, then a third
Type a base side of 9 and a height of 12 in meters above, and you'll get the matching liters and gallons at once.
Units and accuracy
Calcowa shows the pyramid volume in liters, US and UK gallons, milliliters, fluid ounces, and cubic mm, cm, m, inches, feet, and yards all at once. Enter the base and height in any supported unit, and you'll get exact conversions, so the result fits architecture, packaging, and school work alike.
| Unit | Best for | Good to know |
|---|---|---|
| Cubic inches (in³) | Models, small pyramids | Default when you enter inches |
| Cubic feet (ft³) | Roofs, tents, large builds | 1 ft³ = 1,728 in³ |
| Liters (L) | Capacity and containers | 1 L = 1,000 mL |
| US gallons (gal) | Tanks and hoppers | 1 US gallon = 3.785 L |
| Cubic yards (yd³) | Soil, fill, big structures | 1 yd³ = 27 ft³ |
Frequently asked questions
Does the volume formula work for any pyramid base?
Yes. V = ⅓ × base area × height works for a square, rectangular, triangular, or any other base. Only the base area changes; the one-third and the height stay the same, so you just swap in the right base shape.
The volume of a pyramid is V = ⅓ × base area × height, whatever the base shape. For a square base, that's ⅓ × side² × height. A pyramid holds exactly one-third of the prism with the same base and height.
For a square-based pyramid, the base area is the side squared, so V = ⅓ × a² × h. With a 6-unit base and a height of 4, that's ⅓ × 36 × 4 = 48 cubic units. Set the base to Square and enter the side and height.
A pyramid and a prism with the same base and height are linked: three identical pyramids fit exactly inside the matching prism. That's where the one-third comes from, so V = ⅓ × base area × height instead of base area × height.
Add the base area to the lateral faces: SA = base area + lateral area. For a square pyramid, the lateral area is 2 × a × slant height, so SA = a² + 2a × l. The calculator shows the surface area and slant height for square and rectangular bases.
The slant height is the distance from the apex down the middle of a triangular face to the base edge, not the vertical height. For a square pyramid it's l = √((a÷2)² + h²), found from the height and half the base, since they make a right triangle.
Rearrange the volume formula: h = 3V ÷ (base area). If you have the slant height instead, the vertical height is h = √(l² − (a÷2)²) for a square base, using the Pythagorean theorem.
A triangular pyramid, or tetrahedron-style pyramid, still uses V = ⅓ × base area × height. Work out the triangular base area first (½ × base × height of the triangle), then multiply by a third of the pyramid's height. Set the base to Triangular in the calculator.
Related calculators
Working with pointed or boxy solids? These geometry tools pair well with the pyramid.
A round pyramid, also a third of its prism.
Rectangular prism volumeThe box a pyramid fills a third of.
Triangular prism volumeAnother prism for comparison.
Need a pyramid volume fast?
Try the calculator above, or browse every shape in the geometry hub.