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Volume with cubes of fractional side lengths

Before, we've found volume by seeing how many cubes with 1-unit side lengths would fit into an object. Find out what happens when we find volume with smaller cubes.

What is volume?

Volume is the amount of 3-dimensional space an object occupies. We measure volume in cubic units.
For example, the rectangular prism below has a volume of 24 cubic units because it is made up of 24 unit cubes.
We can also find the volume of a rectangular prism by multiplying the side lengths.
4, start text, u, n, i, t, s, end text, dot, 2, start text, u, n, i, t, s, end text, dot, 3, start text, u, n, i, t, s, end text, equals, 24, start text, c, u, b, i, c, space, u, n, i, t, s, end text
That works well when we can fill the prism completely with unit cubes. How could we find volume when a prism has fractional side lengths and spaces too small to fill with unit cubes?

Filling a unit cube with smaller cubes

Let's try starting with smaller cubes.
1.1
This is a cubic centimeter because each of its sides is 1, start text, c, m, end text long.
How many dice with edge lengths of start fraction, 1, divided by, 2, end fraction, start text, c, m, end text do we need to fill the cubic centimeter?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
dice

1.2
What is the volume, in cubic centimeters, of a die with edge lengths of start fraction, 1, divided by, 2, end fraction, start text, c, m, end text?
  • Your answer should be
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
cubic centimeters

Filling a rectangular prism with smaller cubes

Let's consider the following rectangular prism.
2.1
What is the volume of the prism?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
start text, c, m, end text, cubed

2.2
Label how many dice with edge lengths of start fraction, 1, divided by, 2, end fraction, start text, c, m, end text would fit across the length, width, and height of the same prism.
Click each dot on the image to select an answer.

2.3
Based on the numbers of dice you found above, how many dice with edge lengths of start fraction, 1, divided by, 2, end fraction, start text, c, m, end text would it take to fill the prism?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
dice

Before continuing, take a minute to tell a friend how you know the number of dice it would take to fill the prism.
2.4
How does the volume of the prism, in cubic centimeters, relate to the number of dice with edge lengths of start fraction, 1, divided by, 2, end fraction, start text, c, m, end text it takes to fill the prism?
The volume of the prism, in cubic centimeters, is
times the number of dice it takes to fill it.

Why is the number of dice different from the volume?
2.5
Suppose we fill the following prism with dice with start fraction, 1, divided by, 2, end fraction, start text, c, m, end text side lengths.
What is the product of the number of dice and the volume per die, and what does that product represent?
Choose 1 answer:
Choose 1 answer:

Finding volume with cubes with fractional side lengths

Now we know 2 different ways to find the volume of a prism with whole number side lengths:
  1. Find the number of cubes of some size that would fit it and multiply by the volume per cube.
  2. Multiply the side lengths.
Either method gives us the same volume. Do you have another method? Tell us about it below in the comments.

Prisms with fractional side lengths

Suppose we fill the following prism with cubes with side lengths of start fraction, 1, divided by, 4, end fraction, start text, c, m, end text.
How would you find the number of cubes that fill the prism?
3.1
How many cubes with side lengths of start fraction, 1, divided by, 4, end fraction, start text, c, m, end text does it take to fill the prism?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
cubes

3.2
What is the volume, in cubic centimeters, of a cube with side lengths of start fraction, 1, divided by, 4, end fraction, start text, c, m, end text?
  • Your answer should be
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
start text, c, m, end text, cubed

3.3
What is the volume of the prism?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
start text, c, m, end text, cubed

3.4
How is the number of cubes with side lengths of start fraction, 1, divided by, 4, end fraction, start text, c, m, end text related to the volume, in cubic centimeters, of the prism?
The number of cubes is
times the cubic centimeters of volume of the prism.

3.5
Evaluate 1, start fraction, 1, divided by, 4, end fraction, start text, c, m, end text, times, 2, start fraction, 1, divided by, 2, end fraction, start text, c, m, end text, times, 3, start text, c, m, end text (the product of the dimensions of the prism).
  • Your answer should be
  • a simplified proper fraction, like 3, slash, 5
  • a mixed number, like 1, space, 3, slash, 4
start text, c, m, end text, cubed

Summary

Both strategies of finding volume work with rectangular prisms with fractional side lengths, too! Describe those 2 strategies to a friend.
Do you have another way of finding volume when the rectangular prism has fractional side lengths? Tell us about it in the comments.

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