Volume of a Pyramid
Calculate the volume of a pyramid using V = (1/3)Bh.
Works for any pyramid shape — square, rectangular, or triangular base.
The Formula
V = (1/3) × B × h
The volume of a pyramid is one-third of the base area multiplied by the height.
This works for any pyramid, regardless of the shape of its base.
Variables
| Symbol | Meaning |
|---|---|
| V | Volume of the pyramid |
| B | Area of the base (calculate using the appropriate area formula) |
| h | Height of the pyramid (measured perpendicular to the base) |
Example 1
Find the volume of a pyramid with a square base of 6 m sides and height 10 m
Base area B = 6 × 6 = 36 m²
V = (1/3) × B × h = (1/3) × 36 × 10
V = (1/3) × 360
V = 120 m³
Example 2
A pyramid has a rectangular base of 8 cm by 5 cm and a height of 12 cm
Base area B = 8 × 5 = 40 cm²
V = (1/3) × 40 × 12 = (1/3) × 480
V = 160 cm³
When to Use It
Use the volume of a pyramid formula when:
- Calculating the volume of pyramid-shaped structures or objects
- Working with architectural or historical pyramid measurements
- Solving 3D geometry problems involving pyramids
- Comparing volumes — a pyramid is always 1/3 the volume of a prism with the same base and height
Key Notes
- Formula: V = (1/3) × B × h: B is the area of the base (any polygon shape), and h is the perpendicular height from the base to the apex. The factor of 1/3 holds for any pyramid, regardless of the base shape.
- Why the factor is 1/3: A cube can be divided into exactly three congruent pyramids (Cavalieri's principle). This is the geometric proof that a pyramid has exactly 1/3 the volume of a prism with the same base and height.
- Cone is a special case: A cone is a pyramid with a circular base. V = (1/3)πr²h. The same 1/3 factor applies because a cone and a cylinder with equal base and height follow the same relationship.
- Height is perpendicular, not slant: The height h in the formula must be the vertical (perpendicular) distance from the base to the apex, not the slant height along the face. Use the Pythagorean theorem to find h from the slant height if needed.
- Great Pyramid example: The Great Pyramid of Giza has a square base of ~230 m × 230 m and an original height of ~146 m. Its volume: V = (1/3) × 230² × 146 ≈ 2,583,283 m³ ≈ 2.58 million cubic meters.