Pyramid Calculator

Calculate the volume of any pyramid type: square, rectangular, triangular, hexagonal, and more. Essential geometry tool for students and professionals.

Square Pyramid
Triangular Pyramid
Rectangular Pyramid
Hexagonal Pyramid

Square Pyramid Formulas:

Volume: V = (1/3) × b² × h

Surface Area: A = b² + 2b × s

Slant Height: s = √(h² + (b/2)²)

Where: b = base length, h = height, s = slant height

units
Length of one side of the square base
units
Perpendicular height from base to apex
Select units for input and results
Number of decimal places in results
Calculating...

Understanding Pyramids

A pyramid is a polyhedron formed by connecting a polygonal base and a point called the apex. Each base edge and apex form a triangle. Pyramids are classified by the shape of their base.

Key Pyramid Properties:

  • Height (h): The perpendicular distance from the base to the apex.
  • Slant Height (s): The distance from the apex to the midpoint of a base edge along the lateral face.
  • Base Area (Ab): The area of the base polygon.
  • Lateral Area (Al): The sum of areas of all lateral (side) faces.
  • Total Surface Area (At): The sum of base area and lateral area.
  • Volume (V): The space enclosed by the pyramid: V = (1/3) × Ab × h

Pyramid Formulas by Type

Pyramid Type Base Area Volume Surface Area
Square Pyramid b² (1/3) × b² × h b² + 2b × s
Triangular Pyramid (√3/4) × b² (1/3) × (√3/4) × b² × h (√3/4) × b² + (3/2) × b × s
Rectangular Pyramid l × w (1/3) × l × w × h l × w + l × sl + w × sw
Hexagonal Pyramid (3√3/2) × a² (1/3) × (3√3/2) × a² × h (3√3/2) × a² + 3 × a × s

Historical Significance

1

Ancient Egypt: The most famous pyramids are the Egyptian pyramids, built as tombs for pharaohs. The Great Pyramid of Giza is one of the Seven Wonders of the Ancient World.

2

Mesoamerica: Pyramids were also built by Mesoamerican civilizations like the Maya and Aztecs, often as temples or ceremonial platforms.

3

Modern Architecture: Pyramid shapes continue to be used in modern architecture, such as the Louvre Pyramid in Paris and the Luxor Hotel in Las Vegas.

Real-World Applications

  • Architecture: Structural design and aesthetic elements
  • Engineering: Stability analysis and load distribution
  • Packaging: Efficient space utilization in packaging design
  • Education: Teaching geometry and spatial reasoning
  • Gaming: 3D modeling and game development

Calculator Features:

  • Supports four pyramid types: square, triangular, rectangular, and hexagonal
  • Calculates all key geometric properties: volume, surface area, slant height, etc.
  • Provides step-by-step calculation details
  • Includes interactive 3D visualization with rotation and zoom
  • Shows pyramid net (unfolded shape) visualization

Frequently Asked Questions

Height (or altitude) is the perpendicular distance from the base to the apex. Slant height is the distance from the apex to the midpoint of a base edge, measured along the lateral face. For a square pyramid with base length b and height h, the slant height is √(h² + (b/2)²).

A pyramid with a circular base is called a cone. While cones share some properties with pyramids (both have a base and an apex), they are technically different geometric shapes. Pyramids have polygonal bases with straight edges, while cones have curved surfaces.

The volume formula V = (1/3) × Base Area × Height can be derived using calculus or geometric decomposition. One intuitive explanation: a pyramid occupies exactly one-third the volume of a prism with the same base and height. This relationship holds true for all pyramids regardless of base shape.

A right pyramid has its apex directly above the centroid of the base. An oblique pyramid has its apex not aligned with the base center. Our calculator assumes right pyramids, which have symmetrical properties and simpler formulas.

The calculations are mathematically precise based on the formulas for each pyramid type. You can control the decimal precision using the "Decimal Places" selector. For triangular and hexagonal pyramids, calculations involve √3 (approximately 1.73205), which is computed to high precision.