how to find the volume of a pyramid

Volume of Pyramids, Cones and Spheres

To do 5 min read 11 min video

Volume of Pyramids, Cones and Spheres

Contents

Volume of Pyramids, Cones and Spheres
Example
- Question
- Sketch and label the pyramid
- Select the correct formula and substitute the given values
- Write the final answer
Example
- Question
- Sketch the base triangle and calculate its area
- Sketch the side triangle and calculate pyramid height $H$
- Calculate the volume of the pyramid
- Write the final answer
Example
- Question
- Find the area of the base
- Calculate the volume
- Write the final answer
Example
- Question
- Use the formula to find the volume
- Write the final answer
Example
- Question
- Calculate the volume of the prism
- Calculate the volume of the pyramid
- Calculate the total volume
Example
- Question
- Calculate the surface area of each exposed face of the pyramid
- Calculate the surface area of each side of the prism
- Calculate the total surface area of the object

Square pyramid		$\begin{array}{rl} \text{Volume}& = \cfrac{1}{3}\times \text{area of base} \times \\ & \text{height of pyramid}\\ & = \cfrac{1}{3}\times {b}^{2}\times H \end{array}$
Triangular pyramid		$\begin{array}{rl} \text{Volume} & = \cfrac{1}{3}\times \text{area of base} \times \\ & \text{height of pyramid}\\ & = \cfrac{1}{3} \times \cfrac{1}{2}bh \times H \end{array}$
Right cone		$\begin{array}{rl} \text{Volume}& = \cfrac{1}{3}\times \text{area of base} \times \\ & \text{height of cone}\\ & = \cfrac{1}{3}\times \pi {r}^{2}\times H \end{array}$
Sphere		$\text{Volume}=\cfrac{4}{3}\pi {r}^{3}$

This video gives an example of calculating the volume of a sphere.

Example

Question

Find the volume of a square pyramid with a height of 3 cm and a side length of 2 cm.

Sketch and label the pyramid

Select the correct formula and substitute the given values

\[V = \cfrac{1}{3}\times {b}^{2}\times H\]

We are given $b=2$ and $H=3$, therefore

\begin{align*} V & = \cfrac{1}{3}\times {2}^{2}\times 3 \\ & = \cfrac{1}{3}\times 12 \\ & = \text{4}\text{ cm$^{3}$} \end{align*}

Write the final answer

The volume of the square pyramid is $\text{4}$ $\text{cm$^{3}$}$.

Example

Question

Find the volume of the following triangular pyramid (correct to 1 decimal place):

Sketch the base triangle and calculate its area

The height of the base triangle (${h}_{b}$) is:

\begin{align*} {8}^{2}& ={4}^{2}+{h}_{b}^{2} \\ \therefore {h}_{b}& =\sqrt{{8}^{2}-{4}^{2}} \\ & = 4\sqrt{3}~\text{cm} \end{align*}

The area of the base triangle is:

\begin{align*} \text{ area of base triangle}& =\cfrac{1}{2}b\times {h}_{b} \\ & =\cfrac{1}{2}\times 8\times 4\sqrt{3} \\ & =16\sqrt{3} ~\text{cm$^{2}$} \end{align*}

Sketch the side triangle and calculate pyramid height $H$

Calculate the volume of the pyramid

\begin{align*} V & = \cfrac{1}{3}\times \cfrac{1}{2}b{h}_{b} \times H \\ & = \cfrac{1}{3} \times 16\sqrt{3}\times \sqrt{130} \\ & = \text{105.3}\text{ cm$^{3}$} \end{align*}

Write the final answer

The volume of the triangular pyramid is $\text{105.3}$ $\text{cm$^{3}$}$.

Example

Question

Find the volume of the following cone (correct to $\text{1}$ decimal place):

Find the area of the base

\begin{align*} \text{area of circle}& =\pi {r}^{2} \\ & = \pi \times {3}^{2} \\ & = 9\pi ~\text{cm$^{2}$} \end{align*}

Calculate the volume

\begin{align*} V & =\cfrac{1}{3}\times \pi {r}^{2}\times H \\ & = \cfrac{1}{3}\times 9\pi \times 11 \\ & = \text{103.7}\text{ cm$^{3}$} \end{align*}

Write the final answer

The volume of the cone is $\text{103.7}$ $\text{cm$^{3}$}$.

Example

Question

Find the volume of the following sphere (correct to 1 decimal place):

Use the formula to find the volume

\begin{align*} \text{volume}& =\cfrac{4}{3}\pi {r}^{3} \\ & =\cfrac{4}{3}\pi {(4)}^{3} \\ & = \text{268.1}\text{ cm$^{3}$} \end{align*}

Write the final answer

The volume of the sphere is $\text{268.1}$ $\text{cm$^{3}$}$.

Example

Question

A triangular pyramid is placed on top of a triangular prism, as shown below. The base of the prism is an equilateral triangle of side length $\text{20}$ $\text{cm}$ and the height of the prism is $\text{42}$ $\text{cm}$. The pyramid has a height of $\text{12}$ $\text{cm}$. Calculate the total volume of the object.

Calculate the volume of the prism

First find the height of the base triangle using the theorem of Pythagoras:

Next find the area of the base triangle:

\begin{align*} \text{area of base triangle} & = \cfrac{1}{2}\times 20\times 10\sqrt{3} \\ & = 100\sqrt{3} ~\text{cm$^{2}$} \end{align*}

Now we can find the volume of the prism:

\begin{align*} \therefore \text{volume of prism} & = \text{area of base triangle}\times \text{height of prism} \\ & = 100\sqrt{3}\times 42 \\ & = \text{4 200}\sqrt{3} ~\text{cm$^{3}$} \end{align*}

Calculate the volume of the pyramid

The area of the base triangle is equal to the area of the base of the pyramid.

\begin{align*} \therefore \text{volume of pyramid} & = \cfrac{1}{3}(\text{area of base})\times H \\ & = \cfrac{1}{3}\times 100\sqrt{3}\times 12 \\ & = 400\sqrt{3} ~\text{cm$^{3}$} \end{align*}

Calculate the total volume

\begin{align*} \text{total volume} & = \text{4 200} \sqrt{3} + 400\sqrt{3} \\ & = \text{4 600} \sqrt{3} \\ & = \text{7 967.4}\text{ cm$^{3}$} \end{align*}

Therefore the total volume of the object is $\text{7 967.4}$ $\text{cm$^{3}$}$.

Example

Question

With the same complex object as in the previous example, you are given the additional information that the slant height ${h}_{s}$ = $\text{13.3}$ $\text{cm}$. Now calculate the total surface area of the object.

Calculate the surface area of each exposed face of the pyramid

\begin{align*} \text{area of one pyramid face}& = \cfrac{1}{2}b\times {h}_{s} \\ & = \cfrac{1}{2}\times 20 \times \text{13.3} \\ & = \text{133}\text{ cm$^{2}$} \end{align*}

Because the base triangle is equilateral, each face has the same base, and therefore the same surface area. Therefore the surface area for each face of the pyramid is $\text{133}$ $\text{cm$^{2}$}$.

Calculate the surface area of each side of the prism

Each side of the prism is a rectangle with base $b = \text{20}\text{ cm}$ and height ${h}_{p} = \text{42}\text{ cm}$.

\begin{align*} \text{area of one prism side}& = b\times {h}_{p} \\ & = 20\times 42 \\ & = \text{840}\text{ cm$^{2}$} \end{align*}

Because the base triangle is equilateral, each side of the prism has the same area. Therefore the surface area for each side of the prism is $\text{840}$ $\text{cm$^{2}$}$.

Calculate the total surface area of the object

\begin{align*} \text{total surface area} =& \text{area of base of prism} + \text{area of sides of prism} + \text{area of exposed faces of pyramid}\\ = & (100\sqrt{3}) + 3(840) + 3(133)\\ =& \text{3 092.2}\text{ cm$^{2}$} \end{align*}

Therefore the total surface area (of the exposed faces) of the object is $\text{3 092.2}$ $\text{cm$^{2}$}$.

This video shows an example of calculating the volume of a complex object.

[Attributions and Licenses]

Tutorial Lessons

Measurements: Area of a Polygon

Right Prisms and Cylinders

Surface Area of Prisms and Cylinders

Volume of Prisms and Cylinders

Right Pyramids, Right Cones and Spheres

Surface Area of Pyramids, Cones and Spheres

Volume of Pyramids, Cones and Spheres

The Effect of Multiplying a Dimension by a Factor of K

Summary and Main Ideas

Right Prisms and Cylinders

Surface Area of Prisms and Cylinders

Volume of Prisms and Cylinders

Right Pyramids, Right Cones and Spheres

Surface Area of Pyramids, Cones and Spheres

Volume of Pyramids, Cones and Spheres