Pyramids have fascinated humans since always: learn how to calculate the volume of a pyramid with our tool.

Here you will learn:

  • What is a pyramid, and their elements;
  • How to calculate the volume of a pyramid;
  • The general formula for the volume of a pyramid.

And much more. Insert the data; we'll give you the result in a second!

What is a pyramid?

A pyramid is a 3D solid composed by:

  • A polygonal base, with
    • nn sides; and
    • nn angles.
  • nn lateral faces.

The sides of the lateral oblique faces connect at the apex of the pyramid.

We can identify different types of pyramids:#

  • Regular pyramids: the base is a regular polygon;
  • Irregular pyramids: the base is any polygon;
  • Oblique pyramids: the apex doesn't lie above the centroid of the base;
  • Right pyramid: the apex is exactly above the centroid of the base.

A pyramid with an infinite number of sides is a cone. You can learn how to calculate the volume of a cone, and see the difference between these calculations and the one for a pyramid.

How to find the volume of a pyramid

To find the volume of a pyramid. one formula fits all:

V=13BhV=\frac{1}{3}\cdot B\cdot h

Where:

  • VV is the volume;
  • BB is the area of the base; and
  • hh is the height of the pyramid.

This formula for the pyramid volume is valid for every right and oblique pyramid. This is how to calculate the volume of a pyramid.

Only for regular pyramids you can find the volume without knowing the surface area: it's enough to know the value of the base's side.

V=n12ha2cot(πn)V= \frac{n}{12} \cdot h \cdot a^2 \cdot \cot{\left(\frac{\pi}{n}\right)}

Where:

  • nn is the number of the sides of the bases; and
  • aa is the side's length.

🙋 Alternatively, you can find the area of the base with our area of regular polygon calculator.

How to calculate the volume of a pyramid with a square base

Let's look in detail at the case of a pyramid with a square base.

The formula for the volume of the pyramid is:

V=13a2hV = \frac{1}{3}\cdot a^2 \cdot h

aa is the side of the base; thus, a2a^2 is the area of the base.

This formula for the volume of a pyramid with a square base is equivalent to the more complex expression:

V=412ha2cot(π4)V= \frac{4}{12} \cdot h \cdot a^2 \cdot \cot{\left(\frac{\pi}{4}\right)}

Since cot(π4)=1\cot{\left(\tfrac{\pi}{4}\right)}=1, you can see how this expression simplifies to the first one.

For example. define the lengths of sides and height:

  • a=4 cma=4\ \text{cm}; and
  • h=12 cmh=12\ \text{cm}.

Calculate the area of the base:

B=a2=42=16 cm2B=a^2=4^2 = 16\ \text{cm}^2

Now calculate the volume of this pyramid with a square base:

v=131612=64 cm3v=\frac{1}{3}\cdot 16\cdot 12 = 64\ \text{cm}^3

How to use our pyramid volume calculator

Use our tool to calculate the volume of a regular pyramid, and not only!

Choose the base type. It can be either:

  • Regular-base area;
  • Irregular-base area; and
  • Known base area.

This choice will change the calculator's behavior: insert the values you know: we will calculate everything else!

🙋 Our calculators work in reverse, too: you can find the height or the base area if you know the volume.

If you click on advanced mode, you'll see more parameters of the pyramids you are interested in:

  • Slanted height (we calculate it with the Pythagorean theorem);
  • Lateral edge length; and
  • Surface area.
Davide Borchia
Base type
Regular-shaped base
Pyramid with a regular base with 4 sides

The volume formula works for both right and oblique pyramids.
Shape of base
square (4-sides)
Height (h)
in
Side length (a)
in
Volume
cu in
Use Advanced mode to show slant height, lateral edge length and surface area, for right pyramids.
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