Volume of a pyramid

The volume of a pyramid can be found if we can work out the area of the base and we have the height of the pyramid. We multiply the base by the height, and then we divide by 3.

When do I use this?

Whenever a task asks how much fits inside a pyramid. The base can be all sorts of shapes, which is the one thing to watch. What volume means in general is explained in 3D shapes: volume and surface area.

The formula

$$V = \frac{A_{\text{base}} \times h}{3}$$

Here $A_{\text{base}}$ is the area of the base and $h$ is the height of the pyramid, straight up from the base to the point.

Remember that the pyramid can have different shapes as its base. That could be a triangle, a quadrilateral or a pentagon. The formula stays the same, but the first step changes: we have to work out the area of whatever the base is before we can use it.

Worked example

Take a pyramid with a square base, $6$ cm by $6$ cm, and a height of $5$ cm.

First the area of the base. A square has area side times side:

$$A_{\text{base}} = 6 \times 6 = 36 \text{ cm}^2$$

Then the volume:

$$V = \frac{36 \times 5}{3} = \frac{180}{3} = 60 \text{ cm}^3$$

So $60$ cubic centimetres fit inside the pyramid.

If the base had been a triangle instead, the only difference would be the first step: the area of a triangle is $\frac{1}{2} \times h \times g$, height times base times a half, as in how to find the area of a triangle. After that it is the same multiply by the height and divide by 3.

The surface area of a pyramid

If the base has sides that are all the same length and angles that are all the same, what we call a regular polygon, we can find the surface area by adding the area of the base to the area of the triangles on the sides of the pyramid. In this formula $n$ is how many triangles there are:

$$S = A_{\text{base}} + n \times \frac{1}{2} \times h \times g$$

The $\frac{1}{2} \times h \times g$ is just the area of one side triangle, with $g$ its base and $h$ its height. Notice that this $h$ is the height of the triangle, measured along the sloping face, not the height of the pyramid.

For the square-based pyramid above, there are $n = 4$ side triangles, each with base $g = 6$ cm. If each side triangle has a height of $7$ cm along the face, then

$$S = 36 + 4 \times \frac{1}{2} \times 7 \times 6 = 36 + 84 = 120 \text{ cm}^2$$

Common mistakes

• Forgetting to divide by 3. Base times height on its own is the box the pyramid would fit inside. The pyramid is that divided by 3.
• Using the wrong height in the surface area. The volume uses the height of the pyramid. The side triangles have their own height along the sloping face, and that is the one that goes into their area.
• Mixing up volume and surface area, or their units. Volume is inside, in cm³. Surface area is the wrapping paper, in cm². If the unit is wrong, we have probably calculated wrong somewhere.

Related

The whole catalogue is in 3D shapes: volume and surface area. The cone is the same idea with a circle as its base, in volume and surface area of a cone. A pyramid with the top cut off is a frustum, in volume of a frustum. The side triangles are explained in triangles.