Volume of a pyramid

Don't you know how to calculate the volume of a pyramid? Use our online calculator and find out without knowing the formula.

Just select the shape of the base of the pyramid and type in the tool some data such as the height of the figure or how long one of the sides of the base is. And remember that you can also calculate the area of a pyramid in this other link that we have just left you.

Formula for calculating the volume of a pyramid

To obtain the volume of a pyramid, the following general formula must be applied:

Formula for calculating the volume of a pyramid

Being:

  • Ab the area of the base
  • h the height of the pyramid

There is a second formula that allows us to get the volume of a pyramid regular, i.e., one that has a regular polygon (square, pentagon, etc.) as its base. It is this:

Formula for calculating the volume of a regular pyramid

Being:

  • N the number of sides of the base polygon
  • L the length of one of the sides of the polygon of the base
  • apb is the length of the apothem of the base
  • h corresponds to the height of the pyramid

The main difficulty lies in the calculation of the area of the base of the pyramid since, depending on its shape, we will have to use one formula or another.

If they do not give us the height, we also have an added difficulty since we will have to calculate it from other data such as the apothem.

To clarify the most frequent cases of the calculation of the volume of a pyramid, let's see some typical solved examples.

Volume triangular pyramid

Formula for the volume of a triangular pyramid

If you are asked to calculating the volume of a triangular pyramidyou can directly apply the formula above these lines. You just have to know how long one of the sides of the base (L) is and the height of the figure.

By example.Let's calculate the volume of one whose base measures 2 meters and has a height of 3 meters:

Volume triangular pyramid = √3/12 - 22 - 3 = √3 = 1,732 m3

Volume quadrangular pyramid

Formula for the volume of a quadrangular pyramid

In case you want to calculating the volume of a quadrangular pyramidThe formula to be used is quite simple since the base, being square, makes the calculations much easier.

We are going to see a exercise solved in which we are given a quadrangular pyramid whose side measures 4 meters and has a height of 6 meters:

Volume of quadrangular pyramid = 1/3 - 42 - 6 = 1/3 - 16 - 6 = 32 m3

Volume pentagonal pyramid

Formula for calculating the volume of a pentagonal pyramid

Yes we are given a pentagonal pyramid and asked to calculate its volumethe calculations become a little more complicated. This usually happens because generally in the problem statement we are only given the side and height of the pyramid, but we are not given the apothem of the base, which forces us to perform some more operation.

For example, we are going to obtain the volume of a pentagonal pyramid whose base edge measures 5 meters and has a height of 10 meters.

The first thing we are going to calculate is the value of the apothem of the pentagon of the base, for this you can enter the link that we have just left and in which we explain how to do it to obtain its value:

apb = 5 m / [2tan(36º)] = 3.44 meters

With the apothem of the pentagonal pyramid Once it is obtained, we can apply the general formula that will give us its volume:

Volume of pentagonal pyramid = 5/6 - 5 - 3,44 - 10 = 143,33 m3

Hexagonal pyramid volume

Formula to get the volume of a hexagonal pyramid

Finally, let's take a look at how to calculate the volume of a pyramid with a hexagonal base. In this case the same thing happens as in the pentagonal pyramid and that is that we need to know how much is the apothem of the base to apply the formula.

Here we explain how to calculate the apothem of a hexagon so we will use that knowledge to solve a exercise in which we are asked to find the volume of a hexagonal pyramid 15 meters high whose base has sides measuring 3 meters.

The first thing we have to do is calculate the value of the apothem of the hexagon at the base of the pyramid. If you still don't know how to do it, remember to click on the link we have left in the previous paragraph:

apb = R√3/2 = 3√3/2 = 2.59 meters

Now we have all the data we need to draw the volume of the hexagonal pyramid applying the general formula:

Hexagonal pyramid volume = 3 - 2,59 - 15 = 116,91 m3

If you have any doubt or you have an exercise that you do not know how to solve, leave us a comment and we will help you. Remember that if you want to obtain the volume of any other pyramid that we have not included here, you can always use the general formula to solve the problem.

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