The volume of an 3D object is the total quantity of space the shape occupies.
The volume of an object is measured in cubic units such as cubic centimeters, cubic inch, cubic foot, cubic meter, etc..
Finding the volume of an object can help us to determine the amount required to fill that object like the amount of water needed to fill a bottle, an aquarium or a water tank.

## Volume of Common Shapes

### Cube

The volume of a cube with length s is calculated by:

\(V = s^3\)

### Cuboid

The volume of a cuboid (rectangular prism) with length \(l\), width \(w\) and height \(h\) is calculated by:

\(V = l w h\)

### Cone

The volume of a cone with height \(h\), slant \(l\) and radius \(r\) is calculated by:

\(V = \cfrac{1}{3} \pi r^2 h\)

### Cylinder

The volume of a cylinder with radius \(r\) and height \(h\) is calculated by:

\(V = \pi r^2 h\)

### Triangular Prism

The volume of a triangular prism is calculated by:

\(V = \cfrac{1}{2} b h l\)

### Sphere

The volume of a sphere is calculated as:

\(V = \cfrac{4}{3} \pi r^3\)

Note that the volume of shapes like prisms and cylinders can be calculated as:

\( V = A_{base}h \)

For example if the area of the base of a pentagonal prism is \(12 \; [\text{cm}^2]\) and the height is \(20 \; [\text{cm}]\), the volume is:

\( V = A_{base}h \)

\( V = (12)(20) \)

\( V = 240 \; [\text{cm}^3]\)

A cubic shaped box is \(50\;[cm]\) by \(50 \;[cm]\) by \(50 \;[cm]\). A shipping crate is packed with \(18\) of these boxes, with no extra space in the crate.
What is the volume of the crate in meters?

Show Answer
First convert the box side length to meters:

\( 50 \; \text{cm} * \cfrac{ 1 \; [\text{m}]}{100 \; [\text{cm}]} = 0.50 \; [\text{m}]\)

The volume of a rectangle prism is calculated by:

\( V = l w h \)

\( V_{box} = (0.50) (0.50) (0.50) = 0.125 \; [\text{m}^3] \)

Since there are \(18\) boxes in the crate, the volume of the crate is:

\( V_{crate} = 18 *(0.125) = 2.25 \; [\text{m}^3] \)

The volume of the crate is \( 2.25 \; [\text{m}^3] \).

Find the volume of a can of soda. The radius of the base is \(4\;[cm]\) and the height is \(13\;[cm]\).

Show Answer
The volume of a cylinder is calculated by:

\(V = \pi r^2 h\)

\(V = \pi (4)^2 (13)\)

\(V = 653.45 \; [\text{cm}^3]\)

The volume of the soda can is \(653.45 \; [\text{cm}^3]\)

A water tank is \(12\; \text{feet}\) long, \(5\; \text{feet}\) wide, and \(9\; \text{feet}\) high. A solid metal box which is \(7\; \text{feet}\) long, \(4\; \text{feet}\) wide, and \(6\; \text{feet}\) high is sitting at the bottom of the tank.
The tank is filled with water. What is the volume of the water in the tank?

Show Answer
To calculate the volume of water, we need to calculate the volume of the water tank and subtract the volume of the metal box.
This is because the metal block displaces the water in the tank as it occupies some volume.

The volume of a rectangle prism is calculated:

\( V = l w h \)

The volume of the tank is :

\(V_{tank} = 12 \cdot 5 \cdot 9 = 540 \; [\text{ft}^3]\)

The volume of the metal box is :

\(V_{box} = 7 \cdot 4 \cdot 6 = 168 \; [\text{ft}^3]\)

The volume of water left in the tank is :

\(V_{water} = 540 - 168 = 372 \; [\text{ft}^3]\)