Volume is used to describe 3-dimensional shapes. This means that in addition to a shape's length and width, we now have a third side to consider - the height. Being able to find the volume of different shapes can be very helpful in the real world because most of the items we see every day are 3-dimensional, such as boxes and cans. Let's look at how this works.
Let's start with the basic unit square.
The unit square is a square that is one unit long and one unit wide. We can find the area of any rectangle by breaking it down into unit squares.
By counting the unit squares, we get the area of 6 units².
When working with volume, we are adding another dimension. This is the unit cube. It has a length of one unit, a width of one unit and a height of one unit.
We can find the volume of a shape by counting the number of unit cubes are in the shape. For, example the rectangular prism below has a length of 3 units, a width of 2 units and height of 1 unit.
Instead of having a rectangle made of unit squares, we now have a rectangular prism made of unit cubes. By counting the cubes, we can see that the volume is 6 cubic units, or 6 units³.
Notice that with volume, the units are always to the 3rd power. This is because volume has 3 dimensions so our units are to the third power.
Let's look at another example.
Find the volume of the cube below.
This cube has a length of 2 units, a width of 2 units and height of 2 units. We can think of the cube as two layers of unit cubes stacked on top of each other. Each layer would look like this:
To find the volume, we count the number of cubes that make up the shape. We can see there are four cubes in this layer, so the volume of this layer is 4 cubic units.
Since the shape is made up of two layers, we multiply the volume of one layer by the number of layers we have.
Total Volume: 4 units³ × 2 layers = 8 units³
The total volume of this cube is 8 cubic units.
Another way to think of volume is as the area of the base times the height.
This means as long as we know the area of the base of a shape and the height, we can find the volume.
Examples:
Josephine wants to know the volume of a cement block. She knows it has a base area of 4 square feet and a height of 8 feet. What is the volume of this tower?
Since we know the area of the base and the height of the block, we can easily find the volume of the cube by multiplying the area of the base by the height.
Volume = Base area × Height
Volume = 4 ft² × 8 ft = 32 ft³
The volume of the block is 32 cubic feet.
Let's try another example.
Kathy wants to find out how much can fit inside a metal can. She knows the height of the can is 3 inches and the area of the base is 5 square inches. What is the volume of the can?
We can find the volume of the cube by multiplying the area of the base by the height.
Volume = Base area × Height
Volume = 5 in² × 3 in = 15 in³
The volume of the can is 15 cubic inches.
When finding the volume of rectangular prisms and cubes, we can use a simple formula to find the volume.
Let's try a few examples.
Volume of Rectangular Prisms |
|||
To find the volume of the rectangular prism, we can use the volume formula: |
|||
The length is 4 inches, the width is 4 inches and the height is 3 inches. We can plug those numbers into the formula. |
|||
Final answer - the volume of the shape is 48 cubic inches. |
|||
Volume of Cubes |
|||
A cube is unique because all the sides are the same length. We can use a simplified volume formula for cubes: |
|||
To find the volume of the given cube, we can multiply the given side length by itself by 3 times. | |||
Final Answer - the volume of the cube is 27 cubic inches. | |||
We might need to calculate the volume of a box in order to figure out how much you can fit inside of it.
Let's try an example:
Lauren is moving out of her apartment and needs to buy some boxes to pack up her things. The store sells three sizes of boxes.
Which box should Lauren buy if she wants the biggest possible box?
To find which box is the biggest, we need to find the volume of each box. We can do that by using the volume formula:
Volume = Length × Width × Height
Box 1: volume = 11 × 15 × 10 = 1650 cubic inches
Box 2: volume = 12 × 14 × 10 = 1680 cubic inches
Box 3: volume = 12 × 15 × 9 = 1620 cubic inches
Final answer - Lauren should buy Box 2 because it has the largest volume.