In this explainer, we will learn how to find volumes of triangular or quadrilateral pyramids and solve problems including real-life situations.

### Definition: Pyramids

Pyramids are three-dimensional geometric shapes, or solid objects, where the *base* is
a polygon (triangle, square, rectangle, pentagon, etc.) and all other sides are triangles
that meet at the *apex* or *vertex*.

A **right pyramid** is a pyramid whose apex lies above the centroid of the base.

A **regular pyramid** is a right pyramid whose base is a regular polygon: all the sides
of the base are of equal length, and all the pyramid’s lateral edges are of equal
length.

The *height* of a pyramid is the distance from the apex to the base.

The *slant height* of a pyramid is the distance measured along a lateral face from the
apex to the base edge. In other words, it is the *height* of the triangle comprising a
lateral face.

Now that we have learned what a pyramid is, let us look at its volume.

Imagine that we can fill a pyramid completely with, say, water. If we poured this water into
a prism of the same base and height as the pyramid, we would observe that the level of water
is exactly at **one-third** of the height of the prism.

This is a general rule for any pyramid.

### The Volume of a Pyramid

The volume of a pyramid is one-third of the volume of the prism of the same base and height:

Let us look at some questions.

### Example 1: Finding the Volume of a Pyramid

Determine the volume of the given pyramid.

### Answer

To find the volume of the given pyramid, we need to find the area of its base (here a rectangle) and its height, which is given as 9 cm.

The area of the rectangular basis is .

By plugging the area of the base and the height into the equation , we find .

The volume of the given pyramid is
72 cm^{3}.

### Example 2: Finding the Volume of a Triangular Pyramid

Determine the volume of the given solid.

### Answer

The given solid is a pyramid (a triangular base and all other faces are triangles as well). To find the volume of the given pyramid, we need to find the area of its base (we can choose whatever face) and its height.

If we choose the triangle at the bottom as base, its area is as follows (note that all faces are right triangles):

, where and are the legs of the right triangle. We find .

The height is 14 cm.

By plugging the area of the base and the height into the equation , we find that, to the nearest ten, the volume of the given solid is .

The volume of the given solid is
924 cm^{3}.

### Example 3: Finding the Volume of a Regular Pyramid given Its Lateral Edge and Slant Height

Find the volume of the following regular pyramid approximating the result to the nearest hundredth.

### Answer

The given pyramid is *regular*, which means that its base is a regular polygon: all
the sides of the base are of equal length. Therefore, the base here is a square.

In addition, a regular pyramid is a right pyramid with lateral edges that are equal in length, here 17 cm.

To find the volume of the given pyramid, we need to find the area of its base and its height.

We are given the slant height (15 cm) and the lateral edge. In a regular pyramid, the slant height meets the base side at its midpoint. Half of the square side and the slant height form on each face right triangles whose hypotenuse is the lateral edge.

We can thus apply the Pythagorean theorem in one of these right triangles to find the side, , of the square in centimeters:

Now that we know the square is of side 16 cm, we can find its area by squaring its side:

We need to find the pyramid’s height. For this, we consider the right triangle inside the pyramid formed by the pyramid’s height and its slant height as hypotenuse. The third side is a segment whose length is half the square’s side.

Applying the Pythagorean theorem in this right triangle gives

By plugging the area of the base and the height into the equation , we find that, to the nearest hundredth, the volume of the given solid is .

The volume of the given pyramid to the nearest hundredth is
1,082.76 cm^{3}.

### Example 4: Finding the Volume of a Regular Triangular Pyramid

Find the volume of the following regular pyramid rounded to the nearset hundredth.

### Answer

In the question, we have a regular pyramid, which means that all the sides of the base (here a triangle) are of equal length. The base of the pyramid is therefore an equilateral triangle of side 14 cm, as indicated on the diagram. The height of the pyramid is also given (17 cm). Remember that the volume of a pyramid is given by

Therefore, we need here to find the area of the triangular base, which is given by

where is the base and is the height of the triangle.

Since the base is an equilateral triangle, we can find its height by applying the Pythagorean theorem in the right triangle formed with the triangle’s height as a leg and one of its sides as a hypothenuse, as shown in the diagram.

We find that that is,

Subtracting 49 from each side, we get

Taking the square root of both sides gives us

Substituting this into our equation for the area of the triangular base, we find that and substituting this into our equation for the volume of the pyramid, we get

Be careful here not to confuse the height of the pyramid with the height of the base. We can now calculate the volume of the pyramid using a calculator, rounding our result to the nearest hundredth. We find

The volume of the pyramid rounded to the nearest hundredth is
480.93 cm^{3}.

### Example 5: Finding the Height of a Pyramid given its Volume and Base Area

Find the height of a regular pyramid whose volume is
196 cm^{3} and base area is
42 cm^{2}.

### Answer

In this question, we are given the volume and the area of the base of a regular pyramid, and we need to find its height.

We know that the three parameters are linked via the equation .

Here, is our unknown, so we need to rearrange our equation to make the subject:

After having checked that the volume of the pyramid and the area of its base are given in units that derive from the same length unit (here centimeters), we can plug their values into the equation

We find

The height of the given pyramid is 14 cm.

### Key Points

- Pyramids are three-dimensional geometric shapes, or solid objects, where the base is a polygon (triangle, square, rectangle, pentagon, etc.) and all other sides are triangles that meet at the apex or vertex.
- A right pyramid is a pyramid whose apex lies above the centroid of the base.
- A regular pyramid is a right pyramid whose base is a regular polygon: all the sides of the base are of equal length, and all the pyramid’s lateral edges are of equal length.
- The volume of a pyramid is one-third of the volume of the prism of the same base and height: