Why is the volume of a pyramid one-third of a prism with the same base and height?

Any prism can be dissected into exactly three pyramids of equal volume. Calculus integration and Cavalieri's principle mathematically prove that a cone or pyramid tapering linearly to an apex has exactly 1/3 the volume of its enclosing prism.

What is the difference between vertical height and slant height?

Vertical height (h) is the perpendicular line segment connecting the apex straight down to the center of the base. Slant height (s) is the distance from the apex down the sloping triangular face to the midpoint of a base edge, forming the hypotenuse of an interior right triangle.

Can this calculator solve pyramids with non-square bases?

Yes! The calculator supports square bases, rectangular bases (which have two different slant heights for length and width), and regular equilateral triangular bases (tetrahedrons).

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Pyramid Calculator

Calculate the volume, total surface area, lateral surface area, slant height, and base area of square, rectangular, and regular triangular right pyramids step by step.

Interactive Pyramid Calculator Workspace

Live Solver

Pyramid Calculation: Calculate the 3D volume, slant height, lateral face area, and total surface area for square, rectangular, and regular triangular right pyramids.

Select Pyramid Base

Base Side Edge (a)

Vertical Height (h)

Pyramid Metrics

Volume (V)96.0000

Total Surface Area138.5280

Lateral Face Area102.5280

Base Area (B)36.0000

Slant Height (s)8.5440

Step-by-Step Derivation

1. Base Area: Calculated as B = 36.0000

2. Slant Height (s): Using Pythagorean relationship along the triangular face = 8.5440

3. Volume (V): Exactly one-third of the enclosing prism: V = (1/3) × Base Area × Height = (1/3) × 36.0000 × 8 = 96.0000

4. Total Surface Area: Base Area + Lateral Face Area = 36.0000 + 102.5280 = 138.5280

Mathematical Formula & Variables

Equation ModelVolume = (1/3) × Base Area × Height, Surface Area = Base Area + Lateral Area

Finds the 3D volume, slant height of triangular faces, lateral area, and total surface area of a pyramid based on base dimensions and vertical height.

Variable Definitions

Symbol	Description
Base Area (B)	Area of the polygonal base (s² for square, l × w for rectangle, or (√3/4)s² for equilateral triangle).
Height (h)	Perpendicular vertical altitude from the apex to the center of the base.
Slant Height (s)	Altitude of a triangular lateral face connecting the apex to the base edge.
Volume (V)	Total enclosed 3D capacity: V = (1/3) × B × h.

How to Use the Pyramid Calculator

Select your pyramid base shape (Square, Rectangular, or Regular Triangular).
Input the base dimensions (such as side length or length and width).
Enter the vertical pyramid height (h) measured perpendicular to the base.
Review the calculated volume, total surface area, lateral face area, and slant height in the output panel.

Practical Example Calculation

Scenario Context: An architect is designing a decorative glass skylight shaped like a right square pyramid with a base side of 6 meters and a vertical height of 4 meters.

Step 1: Identify given values: Base side (a) = 6 m, Vertical height (h) = 4 m.

Step 2: Calculate Base Area (B): B = a² = 6² = 36 square meters.

Step 3: Calculate Slant Height (s): s = √(h² + (a/2)²) = √(4² + 3²) = √(16 + 9) = √25 = 5 meters.

Step 4: Calculate Lateral Surface Area (LA): LA = 4 × (0.5 × a × s) = 4 × (0.5 × 6 × 5) = 60 square meters.

Step 5: Calculate Total Surface Area (TA): TA = Base Area + LA = 36 + 60 = 96 square meters.

Step 6: Calculate Volume (V): V = (1/3) × B × h = (1/3) × 36 × 4 = 48 cubic meters.

The square pyramid skylight has a volume of 48 m³, a total surface area of 96 m², and a slant height of 5 m.

Frequently Asked Questions

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