Area with Fractional Side Lengths

Finding the area of a rectangle with fractional side lengths works exactly the same way as finding the area with whole numbers. You use the standard area formula:

$A = l \times w$

where $A$ is the Area, $l$ is the length, and $w$ is the width. When the dimensions are fractions or mixed numbers, you just multiply the fractions to find the total area.

Multiplying Proper Fractions for Area

To find the area when both sides are proper fractions, simply multiply the numerators together and the denominators together. Always remember to simplify your final answer and include square units!

Example: Find the area of a rectangle that is $\frac{3}{4}\text{ ft}$ by $\frac{2}{3}\text{ ft}$.

1. Set up the equation: $A = \frac{3}{4} \times \frac{2}{3}$
2. Multiply the numerators ($3 \times 2 = 6$) and denominators ($4 \times 3 = 12$): $A = \frac{6}{12}$
3. Simplify the fraction: $A = \frac{1}{2}\text{ sq ft}$

Visualizing with Area Models

We can use an area model to see why fraction multiplication works.

Example: A tile is $\frac{1}{2}\text{ m}$ by $\frac{1}{3}\text{ m}$. What is its area?

Imagine a full $1\text{ m}$ by $1\text{ m}$ square.

• If you split the width into halves (2 sections) and the length into thirds (3 sections), the whole square is divided into $2 \times 3 = 6$ smaller equal pieces.
• Your tile takes up exactly $1$ of those halves and $1$ of those thirds, which is $1$ small piece out of the $6$.

Using the formula: $A = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\text{ sq m}$

Area with Mixed Numbers

If the rectangle's dimensions are mixed numbers, you must first convert them into improper fractions before multiplying.

Example: Find the area of a rectangle $2\frac{1}{2}\text{ cm}$ by $1\frac{1}{4}\text{ cm}$.

1. Convert the mixed numbers to improper fractions:
   • $2\frac{1}{2} = \frac{5}{2}$
   • $1\frac{1}{4} = \frac{5}{4}$
2. Multiply the improper fractions: $A = \frac{5}{2} \times \frac{5}{4} = \frac{25}{8}$
3. Convert back to a mixed number (optional but helpful): $A = 3\frac{1}{8}\text{ sq cm}$