Lesson 11
Volume of Prisms
11.1: A Box of Cubes

How many cubes with an edge length of 1 inch fill this box?
 If the cubes had an edge length of 2 inches, would you need more or fewer cubes to fill the box? Explain your reasoning.
 If the cubes had an edge length of \(\frac 12\) inch, would you need more or fewer cubes to fill the box? Explain your reasoning.
11.2: Volumes of Cubes and Prisms
Use cubes or the applet to help you answer the following questions.

Here is a drawing of a cube with edge lengths of 1 inch.
 How many cubes with edge lengths of \(\frac12\) inch are needed to fill this cube?
 What is the volume, in cubic inches, of a cube with edge lengths of \(\frac12\) inch? Explain or show your reasoning.
 Four cubes are piled in a single stack to make a prism. Each cube has an edge length of \(\frac12\) inch. Sketch the prism, and find its volume in cubic inches.

Use cubes with an edge length of \(\frac12\) inch to build prisms with the lengths, widths, and heights shown in the table.

For each prism, record in the table how many \(\frac12\)inch cubes can be packed into the prism and the volume of the prism.
prism
length (in)prism
width (in)prism
height (in)number of \(\frac12\)inch
cubes in prismvolume of
prism (in^{3})\(\frac12\) \(\frac12\) \(\frac12\) 1 1 \(\frac12\) 2 1 \(\frac12\) 2 2 1 4 2 \(\frac32\) 5 4 2 5 4 \(2\frac12\)  Examine the values in the table. What do you notice about the relationship between the edge lengths of each prism and its volume?

 What is the volume of a rectangular prism that is \(1\frac12\) inches by \(2\frac14\) inches by 4 inches? Show your reasoning.
A unit fraction has a 1 in the numerator.

These are unit fractions: \(\frac13, \frac{1}{100}, \frac11\).

These are not unit fractions: \(\frac29, \frac81, 2\frac15\).

Find three unit fractions whose sum is \(\frac12\). An example is: \( \frac18 + \frac18 + \frac14 = \frac12\) How many examples like this can you find?

Find a box whose surface area in square units equals its volume in cubic units. How many like this can you find?
11.3: Cubes with Fractional Edge Lengths
 Diego says that 108 cubes with an edge length of \(\frac13\) inch are needed to fill a rectangular prism that is 3 inches by 1 inch by \(1\frac13\) inch.

Explain or show how this is true. If you get stuck, consider drawing a diagram.
 What is the volume, in cubic inches, of the rectangular prism? Explain or show your reasoning.


Lin and Noah are packing small cubes into a larger cube with an edge length of \(1\frac12\) inches. Lin is using cubes with an edge length of \(\frac12\) inch, and Noah is using cubes with an edge length of \(\frac14\) inch.
 Who would need more cubes to fill the \(1\frac12\)inch cube? Be prepared to explain your reasoning.
 If Lin and Noah each use their small cubes to find the volume of the larger \(1\frac12\)inch cube, will they get the same answer? Explain or show your reasoning.
11.4: Fish Tank and Baking Pan

A nature center has a fish tank in the shape of a rectangular prism. The tank is 10 feet long, \(8\frac14\) feet wide, and 6 feet tall.

What is the volume of the tank in cubic feet? Explain or show your reasoning.
 The nature center's caretaker filled \(\frac45\) of the tank with water. What was the volume of the water in the tank, in cubic feet? What was the height of the water in the tank? Explain or show your reasoning.

Another day, the tank was filled with 330 cubic feet of water. The height of the water was what fraction of the height of the tank? Show your reasoning.


Clare’s recipe for banana bread won’t fit in her favorite pan. The pan is \(8\frac12\) inches by 11 inches by 2 inches. The batter fills the pan to the very top, and when baking, the batter spills over the sides. To avoid spills, there should be about an inch between the top of the batter and the rim of the pan.
Clare has another pan that is 9 inches by 9 inches by \(2\frac12\) inches. If she uses this pan, will the batter spill over during baking?
 Find the area of a rectangle with side lengths \(\frac12\) and \(\frac23\).
 Find the volume of a rectangular prism with side lengths \(\frac12\), \(\frac23\), and \(\frac34\).
 What do you think happens if we keep multiplying fractions \(\frac12\boldcdot \frac23\boldcdot \frac34\boldcdot \frac45\boldcdot \frac56...\)?
 Find the area of a rectangle with side lengths \(\frac11\) and \(\frac21\).
 Find the volume of a rectangular prism with side lengths \(\frac11\), \(\frac21\), and \(\frac13\).
 What do you think happens if we keep multiplying fractions \(\frac11\boldcdot \frac21 \boldcdot \frac13\boldcdot \frac41\boldcdot \frac15...\)?
Summary
If a rectangular prism has edge lengths of 2 units, 3 units, and 5 units, we can think of it as 2 layers of unit cubes, with each layer having \((3 \boldcdot 5)\) unit cubes in it. So the volume, in cubic units, is: \(\displaystyle 2\boldcdot 3\boldcdot 5\)
To find the volume of a rectangular prism with fractional edge lengths, we can think of it as being built of cubes that have a unit fraction for their edge length. For instance, if we build a prism that is \(\frac12\)inch tall, \(\frac32\)inch wide, and 4 inches long using cubes with a \(\frac12\)inch edge length, we would have:
 A height of 1 cube, because \(1 \boldcdot \frac 12 = \frac12\).
 A width of 3 cubes, because \(3 \boldcdot \frac 12 = \frac32\).
 A length of 8 cubes, because \(8 \boldcdot \frac 12 = 4\).
The volume of the prism would be \(1 \boldcdot 3 \boldcdot 8\), or 24 cubic units. How do we find its volume in cubic inches? We know that each cube with a \(\frac12\)inch edge length has a volume of \(\frac 18\) cubic inch, because \(\frac 12 \boldcdot \frac 12 \boldcdot \frac 12 = \frac18\). Since the prism is built using 24 of these cubes, its volume, in cubic inches, would then be \(24 \boldcdot \frac 18\), or 3 cubic inches.
The volume of the prism, in cubic inches, can also be found by multiplying the fractional edge lengths in inches: \(\frac 12 \boldcdot \frac 32 \boldcdot 4 = 3\)
If a rectangular prism has edge lengths \(a\) units, \(b\) units, and \(c\) units, the volume is the product of \(a\), \(b\), and \(c\). \(\displaystyle V = a \boldcdot b \boldcdot c\)
This means that if we know the volume and two edge lengths, we can divide to find the third edge length.
Suppose the volume of a rectangular prism is \(400\frac12\) cm^{3}, one edge length is \(\frac{11}{2}\) cm, another is \(6\) cm, and the third edge length is unknown. We can write a multiplication equation to represent the situation: \(\displaystyle \frac{11}{2} \boldcdot 6 \boldcdot {?} = 400\frac12\)
We can find the third edge length by dividing: \(\displaystyle 400\frac12 \div \left( \frac{11}{2} \boldcdot 6 \right) = {?}\)