Volume of Square Pyramid - Math Steps, Examples & Questions

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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.

What is volume of a square pyramid?

Common Core State Standards

How does this relate to $7$ th grade math?

Grade 7: Geometry (7.G.B.6)
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.

[FREE] Volume Worksheet (Grade 6 to 8)

Use this quiz to check your grade 6 to 8 students’ understanding of volume. 10+ questions with answers covering a range of 6th, 7th and 8th grade volume topics to identify areas of strength and support!

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[FREE] Volume Worksheet (Grade 6 to 8)

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How to calculate the volume of a square pyramid

In order to calculate the volume of a square pyramid:

Calculate the area of the base.
Substitute values into the formula and solve.
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.

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.

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

$80^2=6400$

Substitute values into the formula and solve.

The volume of a square based pyramid can be found by using the formula. But the height is given in a different unit of measurement to the side of the base, $2.5\mathrm{~m}=250\mathrm{~cm.}$

\begin{aligned}V&=\cfrac{1}{3} \, Bh \\\\ V&=\cfrac{1}{3}\times{6400}\times{250} \\\\ V&=533 333.33 \end{aligned}

Write the answer, including the units.

The volume is $533~000\mathrm{~cm}^{3}$ (to $3 \, sf$ ). This can be also written as $0.533\mathrm{~m}^3.$

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.

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

$6^2=36$

Substitute values into the formula and solve.

The volume of a square based pyramid can be found by using the formula. The perpendicular height is not given, but can be found using Pythagoras’ Theorem. The slant height of the pyramid $8\mathrm{~cm}$ is the hypotenuse.

$h=\sqrt{8^{2}-3^{2}}=\sqrt{55}$

Now you have $B=36,~h=\sqrt{55}.$

\begin{aligned}V&=\cfrac{1}{3} \, Bh \\\\ V&=\cfrac{1}{3}\times{36}\times\sqrt{55} \\\\ V&=12\sqrt{55} \\\\ V&=88.9943… \end{aligned}

Note that the base of one right-angled triangle is half the width of the square, as the face of the pyramid is isosceles, so the vertical height of the triangle meets the base at its midpoint.

Write the answer, including the units.

The volume is $89.0\mathrm{~cm}^3$ (to $3 \, sf$ ).

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.

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

$14^{2}=196$

Substitute values into the formula and solve.

The volume of a square based pyramid can be found by using the formula. The perpendicular height is unknown and the lateral edge length has been given.

You will need to apply Pythagoras’ Theorem twice to find the perpendicular height. The value $7$ comes from using the midpoint of the base edge to make right-angled triangles.

First, find the slant length from the lateral edge length.

$x=\sqrt{18^{2}-7^{2}}=5\sqrt{11}$

Then, find the perpendicular height of the square pyramid.

$h=\sqrt{\left(5\sqrt{11}\right)^{2}-7^{2}}=\sqrt{226}$

Now you have $B=196,~h=\sqrt{226}.$

\begin{aligned}V&=\cfrac{1}{3} \, Bh \\\\ V&=\cfrac{1}{3}\times{196}\times\sqrt{226} \\\\ V&=982.175… \end{aligned}

Write the answer, including the units.

The volume is $982\mathrm{~cm}^{3}$ (to $3 \, sf$ ).

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.

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

$6.5^{2}=42.25$

Substitute values into the formula and solve.

The formula for the volume of the square based pyramid can be used to find the perpendicular height, $h.$

\begin{aligned}V&=\cfrac{1}{3} \, Bh \\\\ 87.6&=\cfrac{1}{3}\times{42.25}\times{h} \\\\ h&=\cfrac{87.6\times{3}}{42.25} \\\\ h&=6.2201… \end{aligned}

Write the answer, including the units.

The perpendicular height $h$ is $6.22\mathrm{~cm}$ (to $3 \, sf$ ).

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.

The area of the base can not be found yet. But, when you find it, you can square root it to find the length of the base (the base edge).

Substitute values into the formula and solve.

The formula for the volume of the square based pyramid can be used to find the base area.

\begin{aligned}V&=\cfrac{1}{3} \, Bh \\\\ 1200&=\cfrac{1}{3}\times{B}\times{24.7} \\\\ B&=\cfrac{1200\times{3}}{24.7} \\\\ B&=145.7489… \end{aligned}

You can then square root the base area to find the length of the base.

$\sqrt{145.7489…}=12.072…$

Write the answer, including the units.

The length of the base is $12.1\mathrm{~m}$ (to $1dp$ ).

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

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 $2484\mathrm{~cm}^3.$

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 $cm^3.$

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

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 $128\mathrm{~cm}^{3}.$

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 $3 \, sf$ ).

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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