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Volume of Pyramid

Do you know that the Great Pyramid of Giza measures about 146.7 m high and 230.6 m in base length? Can you imagine how many cubes of sugar measuring 1 m3 would be needed to fill the Great Pyramid of Giza? Herein, you shall be learning about how this can be calculated through the knowledge of the volume of pyramids.

What is a pyramid?

Pyramids are 3-dimensional objects with triangular sides or surfaces that meet at a tip called an apex. The name 'pyramid' often brings to mind the Pyramids of Egypt, which is one of the seven wonders of the world.

In geometry, a pyramid is a polyhedron obtained connecting a polygonal base to a point, called the apex.

Types of pyramids

Pyramids are of various types depending on the shape of their base. A pyramid with a triangular base is called a triangular pyramid, and a rectangular-based pyramid is known as a rectangular pyramid. The sides of a pyramid are triangular and they emerge from its base. They all meet at a point called the apex.

An image showing the various types of pyramids, Njoku - StudySmarter Originals

What is the volume of a pyramid?

You may be wondering how many blocks of sand can make up the Egyptian pyramids. The volume of a pyramid is the space enclosed by its faces. Generally, the volume of a pyramid is a-third of its corresponding prism. Its corresponding prism has the same base shape, base dimensions and height. Thus, the general formula for calculating the volume of a pyramid is,

$V=\frac{1}{3}×bh$

where,

V is the volume of the pyramid

b is the base area of pyramid

h is the height of pyramid

Note that this is the general formula for the volume of all pyramids. Differences in the formulas are based on the shape of the base of the pyramid.

Volume of rectangular pyramids

The volume of rectangular pyramids can be found by multiplying a third of the rectangular base area by the height of the pyramid. Therefore:

$Volumeofrec\mathrm{tan}gularpyramid=\frac{1}{3}×baseArea×height\phantom{\rule{0ex}{0ex}}Basearea=length×breadth\phantom{\rule{0ex}{0ex}}Volume=\frac{1}{3}×l×b×h$

where;

l is the length of the base

b is the breadth of the base

h is the height of the pyramid

An illustration of the sides of a rectangular pyramid, Njoku - StudySmarter Originals

This means that the volume of a rectangular pyramid is a third of the corresponding rectangular prism.

Volume of square-base pyramids

A square base pyramid is a pyramid whose base is a square. The volume of square-based pyramids can be gotten by multiplying one-third of the square base area by the height of the pyramid. Therefore:

$Volumeofsquarebasepyramid=\frac{1}{3}×baseArea×height\phantom{\rule{0ex}{0ex}}Basearea=lengt{h}^{2}\phantom{\rule{0ex}{0ex}}Volume=\frac{1}{3}×{l}^{2}×h$

where;

l is the length of the square base

h is the height of the pyramid

An illustration of the sides of a square base pyramid, Njoku - StudySmarter Originals

Volume of triangular-based pyramids

The volume of triangular base pyramids can be obtained by multiplying one-third of the triangular base area by the height of the pyramid. Therefore:

$Volumeoftriangularbasepyramid=\frac{1}{3}×baseArea×height\phantom{\rule{0ex}{0ex}}Basearea=\frac{1}{2}×baselength×heightoftriangle\phantom{\rule{0ex}{0ex}}Volume=\frac{1}{3}×\frac{1}{2}×b×{h}_{triangle}×{h}_{pyramid}\phantom{\rule{0ex}{0ex}}V=\frac{1}{6}×b×{h}_{triangle}×{h}_{pyramid}$

where;

l is the length of the base

b is the triangular base length

htriangle is the height of the triangular base

hpyramid is the height of the pyramid

An illustration of the sides of a triangular pyramid, Njoku - StudySmarter Originals

Volume of hexagonal pyramids

The volume of hexagonal base pyramids can be gotten by multiplying one-third of the hexagonal base area by the height of the pyramid. Therefore:

$Volumeoftriangularbasepyramid=\frac{1}{3}×baseArea×height\phantom{\rule{0ex}{0ex}}Basearea=\frac{3\sqrt{3}}{2}×lengt{h}^{2}\phantom{\rule{0ex}{0ex}}Volume=\frac{1}{3}×\frac{3\sqrt{3}}{2}×{l}^{2}×h\phantom{\rule{0ex}{0ex}}Volume=\frac{\sqrt{3}}{2}×{l}^{2}×h$

An illustration of the sides of a hexagonal pyramid, Njoku - StudySmarter Originals

A pyramid of height 15ft has a square base of 12 ft. Determine the volume of the pyramid.

Solution

$Volumeofsquarebasepyramid=\frac{1}{3}×{l}^{2}×h\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}l=12ft\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}h=15ft\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}V=\frac{1}{3}×{12}^{2}×15\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}V=5×144\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}V=720f{t}^{3}$

Calculate the volume of the figure below:

Solution

$Thevolumeofthefigure=volumeofrectangularpyramid+volumeofrectangularprism\phantom{\rule{0ex}{0ex}}Volumeofrectanglarpyramid=\frac{1}{3}×l×b×h\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}l=45cm\phantom{\rule{0ex}{0ex}}b=20cm\phantom{\rule{0ex}{0ex}}h=50cm\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Volumeofrectanglarpyramid=\frac{1}{3}×45×20×50\phantom{\rule{0ex}{0ex}}Volumeofrectanglarpyramid=15000c{m}^{3}\phantom{\rule{0ex}{0ex}}Volumeofrectangularprism=l×b×h\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}l=45cm\phantom{\rule{0ex}{0ex}}b=20cm\phantom{\rule{0ex}{0ex}}h=40cm\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Volumeofrectangularprism=45×20×40\phantom{\rule{0ex}{0ex}}Volumeofrectangularprism=36000c{m}^{3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Thevolumeofthefigure=volumeofrectangularpyramid+volumeofrectangularprism\phantom{\rule{0ex}{0ex}}Thevolumeofthefigure=15000+36000\phantom{\rule{0ex}{0ex}}Thevolumeofthefigure=51000c{m}^{3}$

A hexagonal pyramid and a triangular pyramid are of the same capacity. If its triangular base has a length of 6 cm and a height of 10 cm, calculate the length of each side of the hexagon when both pyramids have the same height.

Solution

The first step is to express the relationship in an equation.

According to the problem, the volume of the triangular pyramid equals the volume of the hexagonal pyramid.

Let bt signify the base area of triangular base and bh represent the base area of hexagonal base.

Then:

$Volumeoftriangularpyramid=Volumeofhexagonalpyramid\phantom{\rule{0ex}{0ex}}\frac{{b}_{t}h}{3}=\frac{{b}_{h}h}{3}$

Multiply both sides of the equation by 3 and divide by h.

$\frac{{b}_{t}h}{3}=\frac{{b}_{h}h}{3}\frac{{b}_{t}h}{3}×\frac{3}{h}=\frac{{b}_{h}h}{3}×\frac{3}{h}{b}_{t}={b}_{h}$

This means that the triangular base and the hexagonal base are of equal area.

Recall that we are required to find the length of each side of the hexagon.

${b}_{t}=\frac{1}{2}×baselength×height\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}baselengthoftriangle=6cm\phantom{\rule{0ex}{0ex}}heightoftriangle=10cm\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{b}_{h}=\frac{3\sqrt{3}}{2}×{l}^{2}$

Where l is the length of the side of a hexagon.

Recall that bt = bh, then;

$\frac{1}{2}×6×10=\frac{3\sqrt{3}}{2}×{l}^{2}\phantom{\rule{0ex}{0ex}}\frac{1}{2}×6×10×\frac{2}{3\sqrt{3}}=\frac{3\sqrt{3}}{2}×{l}^{2}×\frac{2}{3\sqrt{3}}\phantom{\rule{0ex}{0ex}}\frac{20}{\sqrt{3}}={l}^{2}$

Take the roots of both sides of the equation.

$\sqrt{{l}^{2}}=\sqrt{11.547}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}l=3.398cm$

Thus each side of the hexagonal base is approximately 3.4 cm.

Volume of Pyramid - Key takeaways

• A pyramid is a 3-dimensional object with triangular sides or surfaces that meet at a tip called an apex
• The various types of pyramids are based on the shape of their base
• The volume of a pyramid is one-third the base area × height

Frequently Asked Questions about Volume of Pyramid

It is the capacity of a pyramid or the space it contains.

The formula used in calculating the volume of a pyramid is one-third the volume of the corresponding prism.

The volume of a pyramid with a square base is calculated by finding the product of one-third of the area of one of the square bases and the height of the pyramid.

The volume of a pyramid with a triangular base is gotten by multiplying one third of the triangular base area by the height of the pyramid.

Final Volume of Pyramid Quiz

Question

What is a pyramid?

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Answer

This is a 3 dimensional shaped object with triangular sides or surfaces that meet at a tip top called an apex.

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Question

What determines the type of pyramid?

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Answer

The shape of the base

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Question

A pyramid is one-third its corresponding prism.

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Answer

Yes

Show question

Question

If the height of an 942 cubic meters square-based pyramid is 30m. Find the base length.

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Answer

5.6m

Show question

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