📏 Enter known values

📐 Accepted units: mm, cm, dm, m, dam, hm, km, in, ft, yd, mi

Formula Reference

render

Worked example

Data
Length
\(b = {\color{#6c5ce7}{4\,\text{cm}}}\)
Width
\(c = {\color{#6c5ce7}{3\,\text{cm}}}\)
Height
\(d = {\color{#6c5ce7}{5\,\text{cm}}}\)
Volume
\(a = \,?\)
Formula
\[a = \frac{{b}\cdot{c}\cdot{d}}{3}\]
Steps
1Substitute the values
\[a = \frac{{\color{#6c5ce7}{4\,\text{cm}}}\cdot{\color{#6c5ce7}{3\,\text{cm}}}\cdot{\color{#6c5ce7}{5\,\text{cm}}}}{3}\]
Each letter is replaced by its value.
2Multiply
\[a = \frac{60\,\text{cm}^{3}}{3}\]
\(4\cdot3 = 12\)\(12\cdot5 = 60\)\(\text{cm} \times \text{cm} = \text{cm}^{2}\)\(\text{cm}^{2} \times \text{cm} = \text{cm}^{3}\)
3Divide
\[a = \frac{60\,\text{cm}^{3}}{3} = 20\,\text{cm}^{3}\]
\(\frac{60}{3} = 20\)
Volume: 20 cm³
Check

If we use the result to calculate Length, we should get the value you entered:

\[b = \frac{3\cdot{a}}{{c}\cdot{d}}\]
\[b = \frac{3\cdot20\,\text{cm}^{3}}{3\,\text{cm}\cdot5\,\text{cm}} = 4\,\text{cm}\]
It gives 4 cm, the same as the value entered.
Calculate the Volume
Please fill in the fields:
Length Width Height
And leave empty
Volume
Calculate the Length
Please fill in the fields:
Volume Width Height
And leave empty
Length
Calculate the Width
Please fill in the fields:
Volume Length Height
And leave empty
Width
Calculate the Height
Please fill in the fields:
Volume Length Width
And leave empty
Height

How to calculate the volume of a pyramid

The volume of any pyramid is one third of the area of its base times its height. This calculator is for pyramids with a rectangular base, or a square base if the length and the width are equal:

\[ V = \frac{\text{Length} \times \text{Width} \times \text{Height}}{3} \]

The height is the perpendicular distance from the apex to the base, not the length of an edge or of a slanted face.

From the volume and two dimensions you get the third one, for example the height:

\[ \text{Height} = \frac{3V}{\text{Length} \times \text{Width}} \]

How to use the calculator

  1. Enter three of the four values: the volume and the length, width and height.
  2. Leave empty the one you want to find.
  3. Press Calculate. For a square base, enter the side in both Length and Width.

Worked examples

1. Base of 6 × 4 m and a height of 9 m

\[ V = \frac{6 \times 4 \times 9}{3} = 72\ \text{m}^3 \]

2. The Great Pyramid of Giza

Its square base measures about 230 m per side and it was originally about 146.6 m tall: \( 230 \times 230 \times 146.6 \div 3 \approx 2{,}585{,}047\ \text{m}^3 \), about 2.6 million m³.

3. Height from the volume

A pyramid with a volume of 100 cm³ and a base of 5 × 6 cm is \( 3 \times 100 \div (5 \times 6) = 10 \) cm tall.

Pyramids with other bases

The rule "base area × height ÷ 3" works for any pyramid; only the base area changes:

  • Triangular base: \( V = \frac{1}{3} \times \frac{b \times h_b}{2} \times H \).
  • Hexagonal base with side \( s \): \( V = \frac{1}{3} \times \frac{3\sqrt{3}}{2} s^2 \times H \).
  • Circular base (a cone): \( V = \frac{1}{3} \pi r^2 H \).

Height from the slant height

In a pyramid with a square base of side \( a \), if you know the slant height \( g \) (from the apex to the middle of a base edge), the height is:

\[ H = \sqrt{g^2 - \left(\frac{a}{2}\right)^2} \]

Example: a base side of 10 and a slant height of 13 give \( H = \sqrt{169 - 25} = 12 \) and \( V = 10 \times 10 \times 12 \div 3 = 400 \).

A pyramid is one third of a box

A pyramid has one third of the volume of a box with the same base and height. A cube can even be cut into three equal pyramids that share one corner as their apex.

When Do You Need to Calculate the Volume of a Pyramid?

🏛️ Architecture and Monuments

Pyramid roofs, skylights and monuments need their volume to plan heating, ventilation and the amount of material.

Size heating and ventilation under pyramid roofs
⛰️ Earthworks and Stockpiles

Material piled on a rectangular base takes a shape close to a pyramid. Its volume estimates how much earth or gravel there is.

Estimate earthworks and piles
📦 Packaging and Displays

Pyramid boxes for gifts, chocolates and tea bags are designed from their volume, and so are pyramid-shaped displays.

Capacity of pyramid packaging
🏺 History and Archaeology

The volume of the Egyptian and Mesoamerican pyramids helps estimate the blocks, labor and time needed to build them.

Context for how ancient builders worked
🍰 Cooking and Molds

Molds for cakes, chocolates and ice in pyramid shapes hold a volume set by their base and height.

Fill molds with the right amount
📚 Geometry Classes

The 1/3 factor and the relation between the pyramid and the prism are classic topics, often combined with the slant height and the Pythagorean theorem.

Practice with volumes and right triangles

Common Mistakes

⚠️ Forgetting to Divide by 3
Common Error: Using length × width × height, which is the volume of the box around the pyramid, three times the real volume.
✅ Solution: V = L × W × H ÷ 3
⚠️ Using the Slant Height or an Edge
Common Error: The slanted faces and the edges are longer than the height. Using them gives a volume that is too large.
✅ Solution: The height goes from the apex straight down to the base
⚠️ Using the Perimeter of the Base
Common Error: Multiplying the perimeter of the base by the height instead of its area.
✅ Solution: Use the base area: length × width
⚠️ Wrong Base Area
Common Error: Using the rectangle formula for a pyramid with a triangular or hexagonal base.
✅ Solution: Calculate the area of the real base, then multiply by the height and divide by 3
⚠️ Mixed Units
Common Error: A base in meters and a height in centimeters. The result is not in any real unit.
✅ Solution: Convert all three dimensions to the same unit
⚠️ Truncated Pyramids
Common Error: Using the pyramid formula for a shape with its top cut off (a frustum), such as a planter.
✅ Solution: Frustum: V = h(A₁ + A₂ + √(A₁A₂)) ÷ 3

Applications by Industry

Architecture & Construction
  • Roofs: Air volume under hip and pyramid roofs
  • Skylights: Volume of pyramid skylights
  • Foundations: Concrete for tapered footings
Civil Engineering & Mining
  • Stockpiles: Volume of material heaped on a rectangular base
  • Excavations: Pits with sloped walls
  • Embankments: Corner sections of earthworks
History & Tourism
  • Egypt: Volume of the pyramids of Giza
  • Mesoamerica: Stepped pyramids such as those of Teotihuacan
  • Museums: Scale models and replicas
Packaging
  • Gift boxes: Capacity of pyramid boxes
  • Tea bags: Volume of pyramid tea bags
  • Displays: Pyramid stands for shops
Science
  • Crystals: Volume of octahedral crystals, made of two pyramids
  • Chemistry: Pyramidal molecular geometry
  • Acoustics: Pyramid foam panels
Education
  • Geometry: Why a pyramid is one third of a prism
  • Calculus: Volume by slices and Cavalieri's principle
  • Exams: Problems with the slant height

External Resources

Other calculators

Volume of a Square Prism
Volume of a Square Prism
Volume of a Cube
Volume of a Cube
Volume of a Cylinder
Volume of a Cylinder
Volume of a Sphere
Volume of a Sphere
Volume of a Rectangular Prism
Volume of a Rectangular Prism
Volume of a Cone
Volume of a Cone
Perimeter of a Triangle
Perimeter of a Triangle
Area of a Rhombus
Area of a Rhombus

Geometry Calculators

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