Square Pyramid Volume Calculator - Volume from Side and Height

The square pyramid volume calculator finds the three-dimensional space inside a regular square pyramid from a base side length and a vertical height. Use it when a geometry problem, a model build, a gravel or sand estimate, or a textbook exercise calls for the volume of a square pyramid and you already have the side length and the perpendicular height of the solid.

• • Classroom geometry: Check V = s^2 * h / 3 homework problems and walk through the base area, height, and final division steps.
• • Model building: Estimate how much plaster, clay, resin, or paper-mache material a square-based pyramid model will hold.
• • Material estimates: Plan cubic volume of sand, gravel, soil, or feed stored in a square-pyramid pile shape.
• • Quick sanity checks: Compare a hand-computed pyramid volume against the calculator before using the value in a larger formula.

A regular square pyramid has a square base with four equal sides, and the apex sits directly above the center of that base. That symmetry is what lets a single side length describe the entire base area and a single vertical height describe the full height of the solid.

The result is a volume, not an area. Keep the input lengths in the same unit, and read the volume in matching cubic units. If you enter inches, the volume is cubic inches; if you enter meters, the volume is cubic meters.

Before the volume step, the Square Area Calculator isolates the s^2 base area calculation when you need to verify that intermediate value on its own.

The calculator squares the side length to get the base area, then multiplies that base area by the vertical height and divides by three.

• s: base side length of the square base
• h: vertical (perpendicular) height from the base to the apex
• base area: s squared, the area of the square base
• one-third factor: comes from V = (1/3) * B * h, the general pyramid volume formula

The general pyramid volume formula is V = (1/3) * B * h, where B is the area of the base and h is the vertical height. For a square base, B is s^2, so the formula simplifies to V = s^2 * h / 3.

The calculator keeps full precision internally and rounds each displayed output to two decimal places so the values are easy to compare against a hand calculation. The base area and base perimeter outputs help you audit the intermediate steps before trusting the final volume.

According to Wolfram MathWorld, the volume of a pyramid is one-third of the base area multiplied by the height.

For a six-sided regular pyramid with the same vertical height, the Volume Hexagonal Pyramid Calculator applies the same V = (1/3) * B * h rule with a different base area formula.

These four ideas decide whether the square pyramid volume formula matches the solid you are measuring.

The most common mistake is using slant height where the calculator asks for vertical height. If you only know the slant height along a triangular face, first convert it to vertical height using the base half-diagonal as a right-triangle leg.

If your solid is a square pyramid frustum (the top is sliced off), this calculator does not apply directly. Truncate the top to find the missing smaller pyramid, then subtract it from the full pyramid to get the frustum volume.

For a square pyramid that also needs outside covering area, the Surface Area Calculator handles the square pyramid shape on a separate page so this volume result stays focused on V.

Use the square pyramid volume calculator with matching length units and enter the vertical height from the base plane, not the slant height along a face.

1. 1 Enter the base side length: Use the length of one side of the square base, not the diagonal across the base.
2. 2 Enter the vertical height: Use the perpendicular distance from the base plane to the apex, measured straight up.
3. 3 Read the volume: Use this output as the cubic capacity inside the regular square pyramid.
4. 4 Check the base area: Use base area to confirm the s^2 step before the height and division are applied.
5. 5 Confirm the base perimeter: Use base perimeter when a separate problem, like fencing or labeling, also needs the base edge total.

Suppose a sandbox is shaped like a regular square pyramid with side length 5 feet and vertical height 6 feet. The calculator returns 50.00 cubic feet of volume, 25.00 square feet of base area, and 20.00 feet of base perimeter. Use the cubic feet value to order sand, then add a small waste allowance for uneven packing. Note that the volume is twice the base area here, which is a quick way to check that the height is exactly twice the one-third factor's denominator.

After calculating cubic inches, cubic feet, or cubic centimeters, the Volume Converter can convert the finished volume into a different cubic unit.

Showing the volume together with the base area and base perimeter makes the result easier to use and easier to check.

• • Fast primary answer: Get the square pyramid volume in one step instead of squaring, multiplying, and dividing by hand.
• • Audited intermediate steps: The base area and base perimeter outputs match the steps a teacher or worksheet would show.
• • Two inputs, three outputs: Enter side length and height, then read V, base area, and base perimeter from one screen.
• • Unit consistency: The volume is returned in cubic units that match the input length unit.
• • Decimal support: Decimal side lengths and heights work for scaled drawings, model parts, and measured piles.

Because the calculator exposes the base area, you can decide whether to reuse that number in another step. A surface area problem might need s^2 for the base, while a fencing estimate might need 4s for the base perimeter.

The outputs also make errors easier to spot. If the base area is wrong, the volume will be wrong. If the base area looks too small, double-check whether you typed the side length or the diagonal across the square.

For other solids such as prisms, cylinders, cones, and spheres, the Volume Calculator groups those shape-specific volume formulas in one place.

The formula is compact, but a few measurement choices affect whether the volume matches the solid you are actually filling.

• • This calculator does not solve for vertical height from slant height. If you only have slant height, first derive vertical height using the base half-diagonal as a right-triangle leg.
• • It does not compute the frustum (truncated pyramid) volume directly. Subtract a smaller top pyramid from the full pyramid if the top of the solid is cut off.
• • The result is a geometric ideal. Real piles of sand, gravel, or grain settle at an angle and lose volume from gaps, so add a packing or waste allowance for material estimates.

The square base area comes from the regular polygon area rule for four equal sides with right angles, which simplifies to s^2. That simplification is why side length alone is enough to define the base.

Volume measures the space inside a three-dimensional solid. For a regular square pyramid, that space equals one-third of a square prism with the same base and height, which is the geometric reason the result is divided by three.

If the base is rectangular but not square, switch to a rectangular prism or rectangular pyramid formula rather than reusing this one with mismatched side lengths.

According to Wolfram MathWorld, the area of a square is the side length squared.

According to OpenStax, the volume of a pyramid equals one-third of the base area times the height of the pyramid.

If the base is round rather than square, the Cone Volume Calculator uses the same V = (1/3) * B * h rule with a circular base area instead of s^2.