Square Pyramid Volume Calculator

Enter the base side and height of a square pyramid to get its volume, surface area and slant height, drawn to scale. Click a dimension on the drawing to change it.

Dimensions

Square Pyramid
Blueprint · Isometric Click an amber dimension to edit it · isometric, not to printed scale

Result

Volume of a square pyramid

Like the cone, a pyramid is one third of the prism around it, so a square pyramid is ⅓·b²h. Each triangular face stands on the base side and rises to the apex along the face slant height, found from the height and half the base.

Editing on the drawing

Click the base side or the height on the figure, enter a value and press Enter to redraw the pyramid to scale.

How to find the volume of a square pyramid

A square pyramid stands a square base up to a single apex. Two measurements set it: the base side and the vertical height. A third length, the slant height up the middle of a face, is needed for surface area.

The formulas

Like a cone, a pyramid holds one third of the prism that would contain it. The face slant height comes from the height and half the base side.

V = ⅓·b²h
s = √(h² + (b⁄2)²)
A = b² + 2b·s

Worked example

Take a square pyramid with base side 74 and height 82.

Step 1 — face slant height

From height and half the base.

s = √(82² + 37²) = √8,093 ≈ 89.96
Step 2 — volume

A third of base area times height.

V = ⅓ × 74² × 82 = ⅓ × 449,032 ≈ 149,677
Step 3 — surface area

Square base plus four triangular faces.

A = 74² + 2 × 74 × 89.96 ≈ 5,476 + 13,314 ≈ 18,790

What else the calculator returns

Alongside volume and surface area you get the base area, the face slant height and the lateral edge — the sloping corner from a base vertex to the apex, which is longer than the face slant.

Where it comes up

Square-pyramid volumes describe roofs and spires, hoppers, monuments and obelisk caps. The same ⅓·b²h applies at any scale, from a paperweight to the Great Pyramid.

Frequently asked questions

How do you find the volume of a square pyramid?

Take one third of the base area times the height: V = ⅓ · b² · h.

What is the difference between slant height and lateral edge?

The slant height runs up the centre of a triangular face to the apex. The lateral edge runs along a corner from a base vertex to the apex and is longer.

How do I find the height from the volume?

Rearrange the formula: h = 3V / b².

Does the surface area include the base?

The total surface area does: the square base b² plus the four triangular faces, 2b·s. The lateral area alone is 2b·s.

Is this the same formula used for the Great Pyramid?

Yes. Any right pyramid on a square base uses V = ⅓ · b² · h, whatever its size.