Problem 65 - Elementary Algebra

Question

The formula for the area of a rectangle in square units is given by \(A=l \cdot w,\) where \(/\) represents the length and \(w\) represents the width. Use this formula to calculate the area of a rectangle with length 12 centimeters and width 3 centimeters.

Step-by-Step Solution

Answer

The area is 36 square centimeters.

Step 1: Identify the Known Values

First, we need to identify the known values from the problem. The length of the rectangle \( l \) is 12 centimeters, and the width \( w \) is 3 centimeters.

Step 2: Write Down the Formula

The formula for the area \( A \) of a rectangle is given by \( A = l \cdot w \). This is the formula we will use to calculate the area.

Step 3: Substitute Known Values into the Formula

Substitute the known values for length and width into the formula. This gives us \( A = 12 \cdot 3 \).

Step 4: Perform the Multiplication

Multiply the length by the width to find the area of the rectangle. So, \( A = 12 \times 3 = 36 \).

Step 5: State the Area

The area of the rectangle is 36 square centimeters.

Key Concepts

Understanding a Rectangle

A rectangle is a four-sided shape with opposite sides that are both equal and parallel. It’s one of the most common shapes you’ll encounter in both geometry and everyday life.
The corners of a rectangle are all right angles, meaning each one forms a 90-degree angle. This unique feature ensures that each pair of opposite sides is the same length.

• Rectangles are often used in architecture, design, and are seen in objects like books, screens, and many more places.
• To better recognize a rectangle, imagine objects such as a door or a table.

In geometry, and especially when calculating areas, the length and width are vital components.

What is Length?

In the context of a rectangle, the length is one of its sides, usually the longer one. It is measured in units such as centimeters, inches, or feet, depending on the context.
Length plays a crucial role when determining the size of a rectangle. It helps us understand how stretched out the rectangle is in one particular direction. Unlike the perimeter, which is a measure around the shape, the length focuses on just one side.

• The notation for representing length is often the letter \( l \).
• In our exercise, this length is given as 12 centimeters.

Length is combined with another dimension, known as width, to calculate the area of the rectangle.

Understanding Width

Width, much like length, is one side of the rectangle, typically considered the shorter side. Together with length, it defines the size and shape of the rectangle.
In many practical applications, width is what gives a two-dimensional object its shape alongside the length. It's important to note:

• Width is often represented by the letter \( w \).
• Width serves to balance the proportions, making a rectangle appear wider or narrower.

In our example, the width is given as 3 centimeters, and together, length and width are used to compute the full area of the rectangle using the formula \( A = l \cdot w \). This multiplication gives us the total space within the rectangle's borders.