Pyramid Calculator
Find the volume of a right pyramid from a square or rectangular base, or from any base area B. The formula is V = (1/3)Bh. For a square or rectangle this page also reports the face slants, the lateral area, and the total surface.
Lateral 60 · total S = 96
- The base is a square of area B = a² = 6² = 36.
- Volume V = (1/3)Bh = (1/3) × 36 × 4 = 48.
- Face slant (apex to a side midpoint) m = √(h² + (a/2)²) = √(16 + 9) = 5. Lateral = 2am = 60. Total S = B + lateral = 96.
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How to Use the Pyramid Calculator
From a square side and height, or from a rectangle’s length, width, and height, this page grows a right pyramid. From any base area B and height it returns only the volume. From a known volume and a square side it recovers the height.
- Choose From side and h, From L, W, and h, From B and h, or From V and side.
- Enter non-negative lengths or a non-negative area. When solving for h from volume, the side must be positive.
- Read the volume (or the recovered height). Square and rectangle modes also report face slants and surface.
A circular base belongs on the Cone Calculator (V = (1/3)πr²h). A pyramid with the tip sliced off belongs on the Frustum Calculator. The matching box — three times this volume — belongs on the Rectangular Prism Calculator. Find a triangular base first on the Triangle Area Calculator, then enter that area as B here, or use the matching prism on the Triangular Prism Calculator (V = Bh, three times this tent). Converting cubic units belongs on the Volume Converter.
Volume and Surface
A pyramid is a polygonal base drawn up to an apex. Volume is one-third of the prism with the same base and height: V = (1/3)Bh. That is the same one-third as a cone, which is the circular-base case. Height is the perpendicular distance from the apex to the plane of the base.
For a right square pyramid the apex sits above the center. A vertical cut from the apex to the midpoint of a side is a right triangle with legs h and a/2, so the face slant is m = √(h² + (a/2)²). Each triangular face has area (1/2)am, and four faces give lateral 2am. Add the square base for the total S = a² + 2am.
A right rectangular pyramid has two pairs of congruent triangular faces. The slants are √(h² + (W/2)²) on the length sides and √(h² + (L/2)²) on the width sides. Lateral area is L times the first slant plus W times the second. From B and h alone you cannot recover those slants, so that mode reports volume only. Solving a square pyramid for height is h = 3V / a².
Pyramid Calculator Examples
Square side 6, height 4
B = 36. V = 48. Face slant m = 5 (the 3-4-5 triangle with legs 4 and 3). Lateral = 60. Total = 96.
Rectangle 8 by 6, height 4
B = 48. V = 64. One pair of faces uses the same 3-4-5 slant 5; the other uses √(16 + 16) ≈ 5.65685. Lateral ≈ 73.941. Total ≈ 121.941.
Base area 36, height 4
V = 48, the same volume as the first example, without assuming the base is a square. Surface stays blank because B does not name the shape.
Volume 48, square side 6
h = 3 × 48 / 36 = 4. The first pyramid recovered from its volume.
Height 0
V = 0. The faces fold onto the base. A pyramid crushed flat is a polygon, not a tent.
A negative side
A length cannot be negative. The calculator rejects −1.
One-Third of the Prism
Cavalieri’s picture again: a pyramid and a cone of the same base area and height have the same volume, and that volume is one-third of the prism (or cylinder) with the same base and height. The square of side 6 and height 4 has pyramid volume 48 and prism volume 144 on the Rectangular Prism Calculator. The cone of base area 36 and height 4 also has volume 48 — enter B = 36 here, or use the Cone Calculator when you have a radius instead.
Do not drop the 1/3 because “it looks like a box.” Do not use a face edge as the height. And do not treat a cone as a pyramid; only a polygonal base lives here.
Common Pyramid Mistakes
Using Bh without the 1/3
That is a prism: a box on the Rectangular Prism Calculator, or a triangular base on the Triangular Prism Calculator. A pyramid is (1/3)Bh.
Using a face edge as the height
Volume wants the perpendicular height from the apex to the base plane. A lateral edge is longer.
Using (1/3)πr²h on a square tent
That is a cone, on the Cone Calculator. A square base is a², not πr².
Asking From B and h for a surface area
B is only an area. Lateral area needs the side lengths. Use the square or rectangle modes.
Treating a frustum as a pyramid
A frustum is a pyramid with the tip sliced off — two parallel bases. That solid belongs on the Frustum Calculator.
Converting liters on this page
V is already in the cube of your input unit. Use the Volume Converter for liters and gallons.
Pyramid Calculator FAQs
What is the volume of a pyramid?
V = (1/3)Bh. For a square of side 6 and height 4, B = 36 and V = 48 cubic units.
What is the surface area of a square pyramid?
Lateral area is 2am with face slant m = √(h² + (a/2)²). Total surface adds the base: a² + 2am. For a = 6 and h = 4 those are 60 and 96.
How do you find the height from the volume?
For a square base, h = 3V / a². The side must be positive.
Is a pyramid one-third of a prism?
Yes, when they share the same base and the same perpendicular height. The 6-by-4 square pyramid is 48; the matching prism is 144.
Does this treat a cone, a frustum, or an oblique pyramid?
A circular base belongs on the Cone Calculator. A cut-off tip belongs on the Frustum Calculator. A leaning apex is out of scope. Convert liters on the Volume Converter.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
Related Calculators
Rectangular Prism CalculatorV = LWh — three times this pyramid, same base and height.
Triangular Prism CalculatorV = Bh — three times this pyramid, same triangular base and height.
Volume ConverterTurn the cubic units into liters or gallons.
Explore More Geometry Tools
Continue with cones, triangles, and solid-geometry tools in the SolverCalc collection.
Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
Sources
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Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.