🔺 Pyramid Volume Calculator
Calculate the volume of various pyramid types
How to Use This Calculator
Select Base Type
Choose the shape of the pyramid base: Square, Triangular, or Rectangular pyramid.
Enter Dimensions
Input the base dimensions (side length for square/triangle, length and width for rectangle) and the perpendicular height of the pyramid.
Get Volume
Click "Calculate Volume" to get the pyramid volume using the formula V = (1/3) × Base Area × Height.
Formula
Volume = (1/3) × Base Area × Height
This formula applies to all pyramids, regardless of base shape
Where:
- Volume = volume of the pyramid
- Base Area = area of the base shape
- Height = perpendicular distance from base to apex
Base Area Formulas:
- Square: Base Area = side²
- Rectangle: Base Area = length × width
- Triangle (Equilateral): Base Area = (side² × √3) / 4
Example 1: Square Pyramid
Base side = 6 units, Height = 10 units
Base Area = 6² = 36 units²
Volume = (1/3) × 36 × 10 = 120 units³
Example 2: Rectangular Pyramid
Base length = 8, width = 5, Height = 12
Base Area = 8 × 5 = 40 units²
Volume = (1/3) × 40 × 12 = 160 units³
Example 3: Triangular Pyramid (Tetrahedron)
Base side = 4 units, Height = 6 units
Base Area = (4² × √3) / 4 = (16 × √3) / 4 ≈ 6.93 units²
Volume = (1/3) × 6.93 × 6 ≈ 13.86 units³
About Pyramid Volume Calculator
A pyramid is a 3D shape with a polygonal base and triangular faces meeting at an apex. This calculator finds the volume of pyramids with square, rectangular, or triangular bases. The volume formula V = (1/3) × Base Area × Height applies to all pyramids, making it a universal calculation method.
When to Use This Calculator
- Architecture: Calculate volumes for pyramid-shaped buildings, roofs, or structures
- Engineering: Determine material volumes for pyramid components
- Mathematics Education: Teach students about 3D geometry and volume calculations
- Construction: Estimate concrete, sand, or other materials needed for pyramid structures
- Design: Plan pyramid shapes with specific volume requirements
- Packaging: Calculate capacity of pyramid-shaped containers
Why Use Our Calculator?
- ✅ Multiple Base Types: Handles square, rectangular, and triangular pyramids
- ✅ Instant Results: Get accurate volume calculations immediately
- ✅ Base Area Included: Shows base area calculation alongside volume
- ✅ Step-by-Step Display: See the formula applied with your values
- ✅ 100% Accurate: Uses precise mathematical formulas
- ✅ Completely Free: No registration required
Understanding Pyramid Volume
The pyramid volume formula comes from the relationship between pyramids and prisms:
- Universal Formula: V = (1/3) × Base Area × Height applies to all pyramids
- Relationship to Prism: A pyramid has exactly 1/3 the volume of a prism with the same base and height
- Height: Must be the perpendicular distance from the base to the apex
- Base Area: Depends on the base shape (square = side², rectangle = length × width, etc.)
Real-World Applications
Architecture: Calculate the volume of a pyramid roof with square base 10 m × 10 m and height 8 m. Base area = 100 m², Volume = (1/3) × 100 × 8 = 266.67 m³. This helps determine insulation or structural material needed.
Construction: A pyramid-shaped concrete structure has rectangular base 6 m × 4 m and height 5 m. Base area = 24 m², Volume = (1/3) × 24 × 5 = 40 m³. This determines the amount of concrete required.
Packaging: A pyramid gift box with square base 15 cm and height 12 cm has base area = 225 cm², Volume = (1/3) × 225 × 12 = 900 cm³. This calculates the box capacity.
Frequently Asked Questions
Why is the volume formula (1/3) × Base Area × Height?
This comes from the mathematical relationship between pyramids and prisms. A pyramid has exactly one-third the volume of a prism with the same base and height. This can be proven using calculus or geometric decomposition methods.
Does this work for any pyramid shape?
Yes! The formula V = (1/3) × Base Area × Height applies to all pyramids, regardless of base shape (triangle, square, rectangle, pentagon, hexagon, etc.). You just need to calculate the base area correctly for the specific shape.
What's the difference between a pyramid and a prism?
A prism has two identical parallel bases connected by rectangular faces. A pyramid has one base and triangular faces meeting at an apex. A pyramid's volume is (1/3) of a prism with the same base and height.
Can I use this for an oblique pyramid?
The height (h) must be the perpendicular distance from the base to the apex. For an oblique pyramid, use the vertical height, not the slant height. The formula still applies as long as you use the correct perpendicular height.
What is a tetrahedron?
A tetrahedron is a triangular pyramid - a pyramid with a triangular base. It has 4 faces (all triangles), 6 edges, and 4 vertices. It's the simplest type of pyramid and the only pyramid with triangular faces only.
How do I find the volume if I only know the slant height?
You need the perpendicular height (h), not the slant height. If you have the slant height, use the Pythagorean theorem to find the perpendicular height: h = √(slant height² - (base/2)²) for a square pyramid, then use h in the volume formula.