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Square Pyramid Calculator

Computes the volume, slant height, lateral edge, base area, lateral and total surface area and face slope in degrees of a right square pyramid from its base side and height.

When to use

You need the volume or surface of a pyramid with a square base and the apex above the centre of the base.

Do not use when: The base is a circle (use cone), a non-square rectangle or triangle, or the apex is off-centre (oblique pyramid).

Formula

volume = base_side² · height / 3; slant_height = √(height² + (base_side/2)²); lateral_edge = √(height² + base_side²/2); lateral_area = 2 · base_side · slant_height; total_surface_area = base_side² + lateral_area; face_slope_deg = atan(2 · height / base_side)

Inputs

Parameter	Type	Unit	Required	Description
`base_side`	number	units	yes	Length of one side of the square base. Range: > 0
`height`	number	units	yes	Perpendicular distance from the base to the apex. Range: > 0

Outputs

Output	Type	Unit	Description
`volume`	number	units³	base_side² × height / 3.
`slant_height`	number	units	√(height² + (base_side / 2)²), height of each triangular face.
`lateral_edge`	number	units	√(height² + base_side² / 2), length of each edge from a base corner to the apex.
`base_area`	number	units²	base_side².
`lateral_area`	number	units²	2 × base_side × slant_height (four triangular faces).
`total_surface_area`	number	units²	base_side² + 2 × base_side × slant_height.
`face_slope_deg`	number	degrees	Angle between a triangular face and the base = atan(height / (base_side / 2)).

Example

Base 6, height 4: {"base_side":6,"height":4} → {"volume":48,"slant_height":5,"lateral_edge":5.831,"base_area":36,"lateral_area":60,"total_surface_area":96,"face_slope_deg":53.1301}

Great Pyramid of Giza (230.4 m base, 146.6 m original height): {"base_side":230.4,"height":146.6} → {"volume":2594046,"slant_height":186.4473,"lateral_area":85914.92,"face_slope_deg":51.8393}

GET https://tttkmbb.com/api/v1/calculate/pyramid?base_side=6&height=4

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/pyramid (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/pyramid · Markdown: https://tttkmbb.com/geometry/pyramid.md · JSON definition: https://tttkmbb.com/geometry/pyramid.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="pyramid"
OpenAPI operationId: calculate_square_pyramid_properties
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Square pyramid (reference)
Wolfram MathWorld – Square Pyramid (reference)

FAQ

Slant height or lateral edge?

The slant height runs from the apex to the midpoint of a base side (the height of a face); the lateral edge runs from the apex to a base corner and is longer.

What about a rectangular base?

Volume is length × width × height / 3, but the two pairs of faces have different slant heights; this calculator only handles square bases.

Related calculators

Cone Calculator — Circular-based counterpart with the same volume formula (base area × height / 3).
Rectangular Prism Calculator — A prism with the same base and height has three times the volume.
Right Triangle Calculator — Slant height and lateral edge come from right triangles with the height as one leg.