Volume of a pyramid calculator
The volume of a pyramid with a 6 cm by 6 cm square base and a height of 4 cm is 48 cm³, a third of the 144 cm³ box with the same base and height.
Volume
- Volume Volume units are the measurement unit cubed: in centimeters the answer is in cubic centimeters.
- 48 cm³
- Formula A third of the box with the same base and height, for the same reason as the cone: three square pyramids fit together into a cube, which is the hands-on proof.
- V = (1/3) × l × w × h = (1/3) × 6 × 6 × 4 = 48
- Measurements used
- l = 6 cm, w = 6 cm, h = 4 cm
- Total surface area A = lw + l√(h² + (w/2)²) + w√(h² + (l/2)²).
- 96 cm²
- Lateral area (four triangles) Two triangles on the length sides with slant height 5 cm, two on the width sides with slant height 5 cm. A right pyramid is assumed: the apex over the center of the base.
- 60 cm²
- In liters 1 liter is 1,000 cubic centimeters, the volume of a 10 cm cube.
- 0.048 L
- In US gallons A US gallon is defined as exactly 231 cubic inches, 3.785411784 liters. An imperial gallon is about 20% larger.
- 0.01268026 gal
- In other cubic units Cubic conversions cube the length factor: a yard is 3 feet, so a cubic yard is 27 cubic feet.
- 0.000048 m³ = 0.001695104 ft³ = 0.0000627816 yd³
| Every length times | Volume | Surface area |
|---|---|---|
| 0.5 | 6 cm³ (×0.125) | 24 cm² (×0.25) |
| 1 | 48 cm³ (×1) | 96 cm² (×1) |
| 2 | 384 cm³ (×8) | 384 cm² (×4) |
| 3 | 1,296 cm³ (×27) | 864 cm² (×9) |
| 10 | 48,000 cm³ (×1,000) | 9,600 cm² (×100) |
- Volume scales with the cube of length and surface area with the square. Double every measurement and the volume goes up 8 times while the surface only goes up 4. That is why large animals keep heat better and why small ice cubes melt faster.
How the Volume Calculator works
The volume of a pyramid is V = (1/3) × B × h, where B is the area of the base and h the vertical height from the base to the apex. For a rectangular base B is length times width, so V = (1/3)lwh. With a 6 cm square base and a height of 4 cm: the base is 36 cm², and (1/3) × 36 × 4 = 48 cm³. A box with the same base and height would hold 144 cm³; the pyramid is exactly a third of it.
The third has a hands-on proof. A cube can be cut into three identical pyramids, each with one face of the cube as its base and the apex at the same far corner, so each is a third of the cube. For a general pyramid the argument is the one for a cone: slices parallel to the base shrink in proportion to the distance from the apex, their areas shrink with the square of it, and adding areas that grow like x² gives a third of the full stack. That is why the 1/3 holds for any base shape: square, rectangle, triangle or circle, where the pyramid becomes a cone.
The mistake to watch for is the height. The formula wants the vertical height, straight down from the apex to the center of the base, not the slant height up the middle of a face and not the length of a sloping edge. In this pyramid the slant height of each face is √(4² + 3²) = 5 cm, and the edge from a base corner to the apex is √(4² + 18) = √34, about 5.831 cm. Use 5 instead of 4 and the volume comes out 60 cm³, 25% too high. When a problem gives the slant height s of a square pyramid with side a, the vertical height is √(s² - (a/2)²).
Units are cubic: centimeters in, cubic centimeters out, and 48 cm³ is 48 mL. The surface area is in square units: the base, 36 cm², plus four triangular faces each ½ × 6 × 5 = 15 cm², for 96 cm² in total. That is where the slant height belongs. The calculator assumes a right pyramid, with the apex over the center of the base, for the surface area; the volume is the same wherever the apex sits, as long as its height above the base is the same.
This calculator takes a rectangular base. For a triangular base (a tetrahedron), work out the triangle's area first and use (1/3) × B × h with that. A pyramid with its top cut off flat is a frustum, V = (h/3)(B₁ + B₂ + √(B₁B₂)) for the two end areas. Its circular version is the frustum shape here. And a cone is a pyramid with a circular base, (1/3)πr²h, by exactly the same reasoning.
Volume Calculator FAQ
What is the formula for the volume of a pyramid?
V = (1/3) × base area × vertical height. A square pyramid with a 6 cm base and 4 cm height is (1/3) × 36 × 4 = 48 cubic centimeters.
Do I use the slant height or the vertical height?
The vertical height, from the apex straight down to the base. The slant height, measured up the middle of a face, is only for the surface area. For a square base of side a, h = √(s² - (a/2)²).
Why is a pyramid one third of a prism?
A cube splits into three identical pyramids that share a corner, and in general the slices of a pyramid shrink with the square of the distance from the apex, which sums to one third of the full stack.
How do I find the volume of a triangular pyramid?
Use the same rule with the triangle's area as the base: V = (1/3) × (½ × b × h of the triangle) × the pyramid's height. This calculator's pyramid shape takes a rectangular base.
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