What is the volume of a pyramid?

V = (1/3) × base area × height. Any pyramid has one-third the volume of a prism with the same base and height.

What is the base area of a square pyramid?

Base area = s², where s is the side length of the square base.

What is the base area of a triangular pyramid?

For an equilateral triangle of side s, base area = (√3/4)s².

What is slant height?

The length along a lateral face from base edge to apex. For a cone: s = √(h² + r²); for a square pyramid: s = √(h² + (a/2)²).

Archimedes showed that a pyramid has 1/3 the volume of a prism with the same base and height (by decomposition or integration).

Where are pyramids used?

Egyptian monuments, Louvre Pyramid, Transamerica building, roofing, tents, and molecular geometry (e.g. methane tetrahedron).

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GEOMETRY3D GeometryMathematics Calculator

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Pyramid — Square & Triangular Bases

Compute volume (⅓×base×height), base area, and slant height for square and triangular pyramids. From the Great Pyramid of Giza to regular tetrahedrons.

Concept Fundamentals

V = (1/3)A_B × h

Volume

A_B = s²

Square Base

A_B = (√3/4)s²

Triangular Base

s = √(h² + (a/2)²)

Slant Height

Did our AI summary help? Let us know.

Volume = (1/3)×base area×height—works for any pyramid Pyramid has 1/3 the volume of a prism with same base and height Slant height used for lateral surface area Great Pyramid of Giza ~2.6 million m³ Regular tetrahedron: V = (√2/12)×edge³

Key quantities

V = (1/3)A_B × h

Volume

Key relation

A_B = s²

Square Base

Key relation

A_B = (√3/4)s²

Triangular Base

Key relation

s = √(h² + (a/2)²)

Slant Height

Key relation

Ready to run the numbers?

Why: Pyramids are among the oldest geometric forms—Egyptian monuments, modern architecture (Louvre, Transamerica), and molecular geometry (tetrahedral methane). Volume = one-third of a prism with same base and height.

How: Select square or triangular base. Enter base dimension (side length) and height. Base area: square = s², equilateral triangle = (√3/4)s². Volume = (1/3)×base area×height. Slant height from Pythagoras.

Volume = (1/3)×base area×height—works for any pyramidPyramid has 1/3 the volume of a prism with same base and height

Sources:Wolfram MathWorld — PyramidKhan Academy — 3D Geometry

Run the calculator when you are ready.

Pyramid Volume CalculatorEnter base type, dimension, and height to compute volume, base area, and slant height

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3D GEOMETRYAncient Wonders

Pyramid — Square & Triangular Bases

Volume = (1/3)×base area×height. From the Great Pyramid to tetrahedrons.

Cube →Cone →

🔺 Sample Pyramids — Click to Load

Pyramid Dimensions

Base Type

Base Side (s)

Height (h)

pyramid_calc.sh

CALCULATED

$ calculate_pyramid --base=square --s=230 --h=146

Volume

2,574,466.6667

Base Area

52,900

Slant Height

185.8521

Height

146

Square Pyramid

s=230, h=146

2,574,466.67 units³

📐 Base: 52900.00↗️ Slant: 185.85

3D Visualization

Property Radar

Property Comparison

Base Area vs Volume

📐 Calculation Breakdown

INPUT

Base Type

Square

Base Dimension (s)

230

Height (h)

146

CALCULATION

Base Area (A_b)

52,900

A_b = s² = 230²

Volume (V)

2,574,466.6667

V = (1/3)A_b×h = (1/3)×52,900×146

Slant Height (s)

185.8521

√(h² + (s/2)²) for square

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

🏛️

Great Pyramid of Giza has volume ~2.6 million m³—one of the largest man-made structures

— Archaeology

📐

Archimedes proved pyramid volume = (1/3)×base×height by comparing to a prism

— Archimedes

🔺

A regular tetrahedron is a triangular pyramid with 4 equilateral faces—a Platonic solid

— Geometry

🇫🇷

Louvre Pyramid in Paris: 35m base, 21.6m height—glass and metal, 1989

— Architecture

📋 Key Takeaways

• Pyramid volume = $\frac{1}{3} A_B \times h$ — one-third of prism with same base and height
• Square base: $A_B = s^2$ . Triangular (equilateral): $A_B = \frac{\sqrt{3}}{4}s^2$
• Slant height = distance from apex to base edge — used for lateral surface area
• Ancient Egyptians and Greeks knew this formula — Euclid formalized it c. 300 BCE

💡 Did You Know?

🏛️Great Pyramid of Giza has volume ~2.6 million m³ — one of the largest man-made structuresSource: Archaeology

📐Archimedes proved pyramid volume = (1/3)×base×height by comparing to a prismSource: Archimedes

🔺A regular tetrahedron is a triangular pyramid with 4 equilateral faces — a Platonic solidSource: Geometry

🇫🇷Louvre Pyramid in Paris: 35m base, 21.6m height — glass and metal, 1989Source: Architecture

📜Moscow Mathematical Papyrus (c. 1850 BCE) shows Egyptians could calculate truncated pyramid volumeSource: History of Math

🔬Tetrahedral molecular geometry (e.g. methane CH₄) uses pyramid geometrySource: Chemistry

📖 How Pyramid Calculations Work

A pyramid has a polygonal base and triangular faces meeting at an apex. Volume = (1/3) × base area × height.

Base Area

Square: $A_B = s^2$ . Equilateral triangle: $A_B = \frac{\sqrt{3}}{4}s^2$ .

Volume

$V = \frac{1}{3} A_B h$ . Works for any pyramid regardless of base shape.

Slant Height

From apex to midpoint of base edge. Square: $s = \sqrt{h^2 + (a/2)^2}$ . Used for lateral surface area.

🎯 Expert Tips

💡 Pyramid vs Prism

Pyramid has 1/3 the volume of a prism with same base and height.

💡 Height vs Slant

Height is perpendicular to base. Slant height is along a face — always ≥ height.

💡 Tetrahedron

Regular tetrahedron: all edges equal. Volume = (√2/12)×edge³.

💡 Units

Use same unit for base and height. Volume in cubic units.

⚖️ Comparison

Feature	This Calculator	Manual	Basic Online
Square & triangular bases	✅	❌	⚠️
Volume, base area, slant height	✅	❌	⚠️
Step-by-step	✅	✅	❌
Famous pyramid examples	✅	❌	❌
3D visualization	✅	❌	❌
AI analysis	✅	❌	❌

❓ FAQ

Height vs slant height?

Height is perpendicular from apex to base. Slant height is along a triangular face from apex to base edge. Volume uses height.

Does V=(1/3)A_B×h work for all pyramids?

Yes — any pyramid (triangle, square, pentagon base, etc.). Just use the correct base area formula.

Pyramid vs cone?

A cone is a pyramid with a circular base. Same formula: V=(1/3)×base area×height.

Truncated pyramid (frustum)?

Different formula: V=(1/3)h(A₁+A₂+√(A₁A₂)). This calculator is for complete pyramids.

Units?

Use same unit for base dimension and height. Volume in cubic units.

How to find height from volume?

Given V and base area A_B: h = 3V/A_B. For square base: h = 3V/s².

What is a regular tetrahedron?

A triangular pyramid with 4 equilateral faces. All edges equal. Volume = (√2/12)×edge³.

📊 Pyramid Stats

1/3

Of prism volume

Triangular faces (square base)

Triangular faces (tri. base)

Apex

📚 Sources

Euclid's Elements ↗

Historical geometry

Wolfram MathWorld — Pyramid ↗

Mathematical reference

Khan Academy — 3D Geometry ↗

Education

Wikipedia — Pyramid (geometry) ↗

General reference

NCTM Standards ↗

Math education

⚠️ Disclaimer: Results are mathematically precise. Historical pyramid dimensions are approximate.

👈 START HERE

⬅️Jump in and explore the concept!

Pyramid — Square & Triangular Bases

Pyramid — Square & Triangular Bases

🔺 Sample Pyramids — Click to Load

Pyramid Dimensions

3D Visualization

Property Radar

Property Comparison

Base Area vs Volume

📐 Calculation Breakdown

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 How Pyramid Calculations Work

Base Area

Volume

Slant Height

🎯 Expert Tips

💡 Pyramid vs Prism

💡 Height vs Slant

💡 Tetrahedron

💡 Units

⚖️ Comparison

❓ FAQ

Height vs slant height?

Does V=(1/3)A_B×h work for all pyramids?

Pyramid vs cone?

Truncated pyramid (frustum)?

Units?

How to find height from volume?

What is a regular tetrahedron?

📊 Pyramid Stats

📚 Sources

Related Calculators

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Tetrahedron Volume Calculator

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We Value Your Privacy

Pyramid — Square & Triangular Bases

Pyramid — Square & Triangular Bases

🔺 Sample Pyramids — Click to Load

Pyramid Dimensions

3D Visualization

Property Radar

Property Comparison

Base Area vs Volume

📐 Calculation Breakdown

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 How Pyramid Calculations Work

Base Area

Volume

Slant Height

🎯 Expert Tips

💡 Pyramid vs Prism

💡 Height vs Slant

💡 Tetrahedron

💡 Units

⚖️ Comparison

❓ FAQ

Height vs slant height?

Does V=(1/3)A_B×h work for all pyramids?

Pyramid vs cone?

Truncated pyramid (frustum)?

Units?

How to find height from volume?

What is a regular tetrahedron?

📊 Pyramid Stats

📚 Sources

Related Mathematics Calculators

Related Calculators

Ellipsoid Volume Calculator

Tetrahedron Volume Calculator

Torus Volume Calculator

Cube Calculator

Sphere Calculator

Cylinder Calculator

We Value Your Privacy