Pyramid Calculator

What is a Square Pyramid?

A square pyramid is a three-dimensional solid with a square base and four triangular faces that meet at a single point called the apex. The most famous examples are the Egyptian pyramids. A right square pyramid has its apex directly above the center of the base.

Pyramid Dimensions

• Base Side (a): The length of one side of the square base
• Height (h): The perpendicular distance from base to apex
• Slant Height (l): The distance from the apex to the midpoint of a base edge
• Apothem: Half of the base side length (a/2)
• Lateral Edge: The edge from apex to a corner of the base

Pyramid Formulas

Slant Height Formula

The slant height runs from the apex to the middle of a base edge:

l = sqrt(h^2 + (a/2)^2)

Lateral Edge Formula

The edge from apex to a corner of the base:

e = sqrt(h^2 + (a * sqrt(2) / 2)^2)
e = sqrt(h^2 + a^2/2)

Volume Formula

Like all pyramids, volume is 1/3 of base area times height:

V = (1/3) * a^2 * h

Base Area Formula

A_base = a^2

Lateral Surface Area

The area of the four triangular faces:

A_lateral = 2 * a * l

Total Surface Area

A_total = A_base + A_lateral
       = a^2 + 2 * a * l

Example Calculations

Example 1: Basic Pyramid

Given base side = 6 units and height = 4 units:

Slant Height = sqrt(4^2 + (6/2)^2)
            = sqrt(16 + 9)
            = sqrt(25)
            = 5 units

Volume = (1/3) * 6^2 * 4
      = (1/3) * 36 * 4
      = 48 cubic units

Base Area = 6^2 = 36 square units

Lateral Area = 2 * 6 * 5 = 60 square units

Total Surface Area = 36 + 60 = 96 square units

Example 2: Great Pyramid Approximation

The Great Pyramid has a base of about 230m and height of 147m:

Slant Height = sqrt(147^2 + 115^2)
            = sqrt(21609 + 13225)
            = sqrt(34834)
            = 186.6 m

Volume = (1/3) * 230^2 * 147
      = (1/3) * 52900 * 147
      = 2,592,100 cubic meters

Applications

Architecture

Pyramidal roofs, monuments, and decorative elements.

Engineering

Calculating material for pyramid-shaped structures and packaging.

Education

Understanding 3D geometry and spatial relationships.

Related Shapes

• Cone: A pyramid with a circular base
• Triangular Pyramid (Tetrahedron): A pyramid with a triangular base
• Frustum: A pyramid with the top cut off
• Prism: A shape with parallel, congruent bases

Interesting Facts

• A pyramid's volume is exactly 1/3 of a prism with the same base and height
• The Great Pyramid of Giza was the tallest structure for over 3,800 years
• Regular tetrahedron (triangular pyramid with equilateral faces) is one of the five Platonic solids