## Volume of Three-Dimensional Shapes

This chapter gives an overview of volume formulas for more complicated three-dimensional objects: triangular prisms and pyramids, cylinders, cones, and spheres.

The volume of a solid is the measure of how much space an object takes up. It is measured by the number of unit cubes it takes to fill up the solid.

Counting the unit cubes in the solid, we have 30 unit cubes, so the volume is: 2 units \begin{align*}\cdot\end{align*}3 units \begin{align*}\cdot\end{align*}5 units = 30 cubic units.

#### Volume of a Prism

A prism is a solid with two congruent polygon bases that are parallel and connected by rectangles. Prisms are named by their base shape.

To find the volume of a prism, find the area of its base and multiply by its height.

\begin{align*}V_{\text{prism}}\end{align*} \begin{align*} = A_{\text{base}}\cdot h\end{align*}

#### Volume of a Cylinder

A cylinder is a three-dimensional solid consisting of two congruent, parallel, circular sides (the bases), joined by a curved surface.

To find the volume of a cylinder, find the area of its circular base and multiply by its height.

\begin{align*}V_{\text{cylinder}}\end{align*} \begin{align*} = \pi r^2 h\end{align*}

#### Volume of a Pyramid

A pyramid is a three dimensional solid with a polygonal base. Each corner of a polygon is attached to a singular vertex, which gives the pyramid its distinctive shape. Each base edge and the vertex form a triangle. Pyramids are named by their base shape.

To find the volume of a pyramid, find the volume of the prism with the same base and divide by three.

\begin{align*}V_{\text{pyramid}} = \frac{A_{\text{base}} \cdot h}{3}\end{align*}

#### Volume of a Cone

A cone is a three-dimensional solid with a circular base whose lateral surface meets at a point called the vertex.

To find the volume of a cone, find the volume of the cylinder with the same base and divide by three.

\begin{align*}V_{\text{cone}}=\frac{\pi r^2 h}{3}\end{align*}

##### Consider:

What is the ratio between the volume of a cylinder and of a cone, having the same radius and height?

The ratio between the volume of a cylinder and a cone with the same radius and height is (1:1 / 3:1 / 1:3 / 2:1).

#### Volume of a Sphere

A sphere is the set of all points in space equidistant from a center point. The distance from the center point to the sphere is called the radius.

The volume of a sphere relies on its radius.

\begin{align*}V_{\text{sphere}}= \frac{4}{3} \pi r^3\end{align*}

#### Volume of a Composite Solid

A composite solid is a solid made up of common geometric solids. The solids that it is made up of are generally prisms, pyramids, cones, cylinders, and spheres.

The volume of a composite solid is the sum of the volumes of the individual solids that make up the composite.

Let's look at some problems where we find the volume.

1. Find the volume of the rectangular prism below.

To find the volume of the prism, you need to find the area of the base and multiply by the height. Note that for a rectangular prism, any face can be the "base", not just the face that appears to be on the bottom.

$Volume = Area \, { }_{\text {base }} \cdot \, height$

$Volume = (length \, \cdot \, width) \cdot \, height$

$Volume = (4) \quad (4) \quad (5)$

$Volume =80 \, in ^{3}$

2. Find the volume of the cone below.

To find the volume of the cone, you need to find the area of the circular base, multiply by the height, and divide by three.

$Volume =\frac{1}{3} \quad Area \, { }_{\text {base }} \, \cdot \, height$

$Volume =\frac{1}{3} \, \pi \, r^{2} \, h$

$Volume =\frac{1}{3} \, \pi \, (7)^{2}\left(1 \frac{4}{2}\right)$

$Volume =196 \, \pi \, cm ^{3}$

3. Find the volume of a sphere with radius 4 cm.

\begin{align*}\text{Volume of Sphere} & = \frac{4}{3} \pi r^3 \\ & = \frac{4}{3} \pi (4)^3 \\ & = \frac{256 \pi}{3} \ \text{cm}^3\end{align*}

\begin{aligned} \text { Volume of Sphere } &=\frac{4}{3} \pi r^{3} \\ &=\frac{4}{3} \pi(4)^{3} \\ &=\frac{256 \pi}{3} cm ^{3} \end{aligned}

#### Examples

##### Example 1

The composite solid below is made of a cube and a square pyramid. The length of each edge of the cube is 12 feet and the overall height of the solid is 22 feet. What is the volume of the solid? Why might you want to know the volume of the solid?

To find the volume of the solid, find the sum of the volumes of the prism (the cube) and the pyramid. Note that since the overall height is 22 feet and the height of the cube is 12 feet, the height of the pyramid must be 10 feet.

\begin{align*} \text{Volume of Prism (Cube)}&=A_{\text{base}} \cdot h \\ &=(12 \cdot 12) (12) \\ &=1728 \ {\text{ft}}^3 \\ \text{Volume of Pyramid} &=\frac{A_{\text{base}} \cdot h}{3}\\ & =\frac{(12 \cdot 12) (10)}{3}\\ & =480 \ \text{ft}^3 \\ \text{Total Volume} & = \text{Volume of Prism (Cube)}+ \text{Volume of Pyramid} \\ &=1728+480 \\ & =2208 \ \text{ft}^3\end{align*}

The volume helps you to know how much the solid will hold. One cubic foot holds about 7.48 gallons of liquid, so

\begin{align*}\text{Gallons of liquid the solid can hold } & = \text{ Volume of solid } \cdot \text{ Number of gallons/cubic foot}\\ & = (2208)(7.48) \\ & = 16,515.84 \ \text{gallons} \\\end{align*}

##### Example 2

The area of the base of the pyramid below is \begin{align*}100 \ \text{cm}^2\end{align*}. The height is 5 cm. What is the volume of the pyramid?

\begin{align*}V_{\text{prism}} & = A_{\text{base}} \cdot h \\ & =(100 \ {\text{cm}}^2) \cdot (5 \ \text{cm}) \\ & =500 \ {\text{cm}}^3\end{align*}

##### Example 3

The volume of a sphere is \begin{align*}\frac{500 \pi}{3} \ \text{in}^3\end{align*}. What is the radius of the sphere?

To find the radius, use the formula: Volume of Sphere = \begin{align*}\frac{4}{3} \pi r^3\end{align*}

\begin{align*}\frac{500 \pi}{3} & = \frac{4}{3} \pi r^3 \\ 4r^3 &=500 \\ r^3 &=125 \\ r &=5 \ \text{in}\end{align*}

##### Example 4

The volume of a square pyramid is \begin{align*}64 \ {\text{in}}^3\end{align*}. The height of the pyramid is three times the length of a side of the base. What is the height of the pyramid?

\begin{aligned} V &=\frac{A_{\text {base }} \cdot h}{3} \\ 64 &=\frac{\left(s^{2}\right)(\not{3} s)}{\not{3}} \\ 64 &=s^{3} \\ \sqrt[3]{64} &=s \\ 4 &=s \end{aligned}

Side \begin{align*}s=4 \ \text{in}\end{align*} and height \begin{align*}h=3(4)=12 \ \text{in}.\end{align*}