# Lesson 14: Fractional Lengths in Triangles and Prisms

Let’s explore area and volume when fractions are involved.

## 14.1: Area of Triangle

Find the area of Triangle A in square centimeters. Show your reasoning. ## 14.2: Bases and Heights of Triangles

1. The area of Triangle B is 8 square units. Find the length of $b$. Show your reasoning. 2. The area of Triangle C is $\frac{54}{5}$ square units. What is the length of $h$? Show your reasoning. ## 14.3: Volumes of Cubes and Prisms

Use the cubes or the applet for the following questions.

GeoGebra Applet fK74qEqW

1. Your teacher will give you a set of cubes with an edge length of $\frac12$ inch. Use them to help you answer the following questions.

1. Here is a drawing of a cube with an edge length of 1 inch. How many cubes with an edge length of $\frac12$ inch are needed to fill this cube? 2. What is the volume, in cubic inches, of a cube with an edge length of $\frac12$ inch? Explain or show your reasoning.
3. 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.

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

1. 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 prism
volume of
prism (cu in)
row 1 $\frac12$ $\frac12$ $\frac12$
row 2 1 1 $\frac12$
row 3 2 1 $\frac12$
row 4 2 2 1
row 5 4 2 $\frac32$
row 6 5 4 2
row 7 5 4 $2\frac12$
2. Analyze the values in the table. What do you notice about the relationship between the edge lengths of each prism and its volume?
3. What is the volume of a rectangular prism that is $1\frac12$ inches by $2\frac14$ inches by 4 inches? Show your reasoning.

## 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: $$2\boldcdot 3\boldcdot 5$$ 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: $$\frac 12 \boldcdot \frac 32 \boldcdot 4 = 3$$