Math resources Geometry Volume

Volume of square pyramid

# Volume of square pyramid

Here you will learn about the volume of a square based pyramid, including how to calculate the volume of a square based pyramid and how to solve problems involving the volume.

Students will first learn about the volume of square pyramids as a part of geometry in 7 th grade. Students will expand on their knowledge throughout high school geometry.

## What is volume of a square pyramid?

The volume of a square pyramid is how much space there is inside the square based pyramid.

A square pyramid is a three dimensional shape made up of flat faces. It has a square base and four triangular faces (triangular sides) which meet at a point, called the apex.

The vertical height is the length from the square base to the apex and is perpendicular to the base of the pyramid.

To calculate the volume of a square based pyramid, use the formula

\text{Volume}=\cfrac{1}{3}\times\text{area of base}\times\text{height}.

The pyramid height is perpendicular to its base.

The formula can also be written as V=\cfrac{1}{3} \, Bh,

where,

• V represents the volume of the pyramid
• B represents the area of its base
• h represents the perpendicular height of the pyramid

If you are not given the perpendicular height, you can use Pythagoras’ theorem to find it from the given slant height or the lateral edge length.

Other types of pyramids include: a rectangular pyramid (with a rectangular base), a triangular pyramid, (with a triangular base), or a hexagonal pyramid (with a hexagonal base).

A tetrahedron is a special type of triangular based pyramid as all of the faces, which are triangular, are congruent.

The square pyramid formula for volume can be used for any pyramid.

## Common Core State Standards

How does this relate to 7 th grade math?

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

• High School: Geometry (HS.G.GMD.A.3)
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

## How to calculate the volume of a square pyramid

In order to calculate the volume of a square pyramid:

1. Calculate the area of the base.
2. Substitute values into the formula and solve.
3. Write the answer, including the units.

## Volume of square pyramid examples

### Example 1: calculating the volume, given the height, of a square pyramid

Find the volume of the square pyramid.

1. Calculate the area of the base.

The area of the base of a square pyramid can be found by squaring the base edge.

9^2=81

2Substitute values into the formula and solve.

The volume of a square pyramid can be found by using the formula V=\cfrac{1}{3} \, Bh. As B=81 and h=13\text{:}

\begin{aligned}V&=\cfrac{1}{3}\times{81}\times{13} \\\\ V&=351 \end{aligned}

3Write the answer, including the units.

The volume 351\mathrm{~m}^{3}.

### Example 2: calculating the volume, given the height, of a square pyramid

Find the volume of the square pyramid. Write your answer to 3 significant figures.

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

### Example 3: calculating the volume of a square pyramid not given perpendicular height

Find the volume of the square pyramid to 3 significant figures. The apex of the pyramid is directly above the center of the square base.

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

### Example 4: calculating the volume of a square pyramid not given perpendicular height

Find the volume of the square pyramid to 3 significant figures. The apex of the pyramid is directly above the center of the square base.

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

### Example 5: calculating the height, h, given the volume

The volume of the square pyramid is 87.6\mathrm{~cm}^{3}. Find the perpendicular height, h. Give your answer to 3 significant figures.

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

### Example 6: calculating the length of the base given the volume

The volume of the square based pyramid is 1200\mathrm{~m}^{3}. Find the base edge. Give your answer correct to 1 decimal place.

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

### Teaching tips for volume of square pyramid

• Have students describe and find real life examples of square pyramids, like the Great Pyramid of Giza. This could also include buildings, or objects around the classroom that are square pyramids.

• While using worksheets has its place, consider assigning students to work together to solve problem-solving tasks related to square pyramids. This allows students to communicate and learn from one another.

### Easy mistakes to make

• Not using the perpendicular height as the height
The height of a square pyramid needs to be the perpendicular height. This is the height that is at a right-angle to the base.

• Not using cubic units
When finding the volume, use cubic units such as cubic centimeters \left(\mathrm{cm}^{3}\right) or cubic meters \left(\mathrm{m}^{3}\right). Square units such as cm^2 or m^2 are for areas.

• Using the wrong volume formula
There are so many different volume formulas in math, so make sure that you use the correct one to find the volume of a pyramid.

\text{Volume}=\cfrac{1}{3}\times \text{area of base} \times \text{height}
Or
V=\cfrac{1}{3}Bh

### Practice volume of square based pyramid questions

1. Work out the volume of this square based pyramid.

2484\mathrm{~cm}^{3}

414\mathrm{~cm}^{3}

7452\mathrm{~cm}^{3}

138\mathrm{~cm}^{2}

Find the volume of a pyramid by using the formula,

V=\cfrac{1}{3} \, Bh.

The base area can be found by squaring the side length of the base square.

18^{2}=324

Substitute the values given into the formula and work out the volume.

V=\cfrac{1}{3}\times{324}\times{23}=2484

The volume is 2484\mathrm{~cm}^3.

2. Find the volume of this square pyramid. Give your answer in cm^3.

4520\mathrm{~cm}^{3}

452\mathrm{~cm}^{3}

45.225\mathrm{~cm}^{3}

45~225\mathrm{~mm}^{3}

Find the volume of a pyramid by using the formula V=\cfrac{1}{3} \, Bh.

The base area can be found by squaring the side length. The side length is in mm so needs converting to cm.

4.5^{2}=20.25

Substitute the values given into the formula and work out the volume.

V=\cfrac{1}{3}\times{20.25}\times{6.7}=45.225

So the volume is 45.225\mathrm{~cm}^{3}.  This is equivalent to 45~225\mathrm{~mm}^{3}, but you have been asked to give the answer in cm^3.

3. Find the volume of this square pyramid. Give your answer in cm^3.

22~248.60\mathrm{~cm}^{3}

33~372.89\mathrm{~cm}^{3}

11~124.29\mathrm{~cm}^{3}

12~000\mathrm{~cm}^{3}

Find the volume of a pyramid by using the formula V=\cfrac{1}{3} \, Bh.

The base area can be found by squaring the length of the side of the square.

30^{2}=900

You have been given the slant length. You can use this to find the perpendicular height of the pyramid using Pythagoras’ Theorem.

h=\sqrt{40^{2}-15^{2}}=5\sqrt{55}

Substitute the values given into the formula and work out the volume.

V=\cfrac{1}{3}\times{900}\times{5}\sqrt{55}=11~124.29…

The volume is 11~124.29\mathrm{~cm}^{3}.

4. Find the volume of this square pyramid. V is directly above B.  Give your answer in cm^3.

640\mathrm{~cm}^{3}

128\mathrm{~cm}^{3}

213\mathrm{~cm}^{3}

384\mathrm{~cm}^{3}

Find the volume of a pyramid by using the formula

V=\cfrac{1}{3} \, Bh.

The base area can be found by squaring the side length.

8^{2}=64

You have been given a slant length. You can use this to find the perpendicular height of the pyramid.

h=\sqrt{10^{2}-8^{2}}=6

Substitute the values given into the formula and work out the volume.

V=\cfrac{1}{3}\times{64}\times{6}=128

The volume is 128\mathrm{~cm}^{3}.

5. The volume of this square based pyramid is 460\mathrm{~cm}^{3}. Find the perpendicular height of the pyramid. Give your answer correct to 3 significant figures.

2.72\mathrm{~cm}

2.73\mathrm{~cm}

8.17\mathrm{~cm}

8.16\mathrm{~cm}

Substitute the values given into volume of a pyramid formula and rearrange it to find the height, h.

\begin{aligned}V&=\cfrac{1}{3} \, Bh \\\\ 460&=\cfrac{1}{3}\times{13^2}\times{h} \\\\ h&=\cfrac{460\times{3}}{13^2} \\\\ h&=8.1656… \end{aligned}

The perpendicular height is 8.17\mathrm{~cm}^{3} (to 3 \, sf ).

6. The volume of this square based pyramid is 89~000\mathrm{~mm}^{3}. Find the side length of the pyramid in centimeters. Give your answer correct to 3 significant figures.

34.2\mathrm{~mm}

342\mathrm{~mm}

58.5\mathrm{~cm}

5.85\mathrm{~cm}

Substitute the values given into the volume of a pyramid formula and rearrange it to find the base area.

The perpendicular height is given in centimeters, so needs converting to millimeters.

\begin{aligned}V&=\cfrac{1}{3} \, Bh \\\\ 89000&=\cfrac{1}{3}\times{B}\times{78} \\\\ B&=\cfrac{89000\times{3}}{78} \\\\ B&=34.23076… \end{aligned}

The area of the base needs to be square rooted to find the side length.

\sqrt{34.23076…}=5.85070…

The perpendicular height is 5.85\mathrm{~cm} (to 3 \, sf ).

## Volume of a square pyramid FAQs

What is a square pyramid?

A square pyramid is a 3D shape with a square base and 4 triangular faces. These faces meet at a common point called the vertex or apex. This is one type of pyramid.

What is a regular pyramid?

A regular pyramid is a 3D shape with a polygonal base (can be a square base, hexagonal base, etc.) and triangular faces that meet at the apex. The height is perpendicular to the base. The side faces and angles are all equal in a regular pyramid.

What is a right square pyramid?

A right square pyramid has lateral edges that are all the same length and the side faces are all equilateral triangles.

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