# 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

1. Here is a drawing of a cube with edge lengths of 1 inch.

1. How many cubes with edge lengths of $$\frac12$$ inch are needed to fill this cube?
2. What is the volume, in cubic inches, of a cube with edge lengths of $$\frac12$$ inch? Explain or show your reasoning.
2. 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.
3. 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 (in3)
$$\frac12$$ $$\frac12$$ $$\frac12$$
1 1 $$\frac12$$
2 1 $$\frac12$$
2 2 1
4 2 $$\frac32$$
5 4 2
5 4 $$2\frac12$$
2. Examine the values in the table. What do you notice about the relationship between the edge lengths of each prism and its volume?
4. 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: $$\frac29, \frac81, 2\frac15$$.

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

2. Find a box whose surface area in square units equals its volume in cubic units. How many like this can you find?

### 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: ​​​​​​$$\frac 12 \boldcdot \frac 32 \boldcdot 4 = 3$$