Problem 64 - Prealgebra

Question

Find the area and perimeter for a rectangle if the length and width are as given below. \(l=40\) yards, \(w=20\) yards

Step-by-Step Solution

Answer

The area is 800 square yards and the perimeter is 120 yards.

Step 1: Understanding the Problem

We need to find two values for the rectangle: the area and the perimeter. We are given the length \( l = 40 \) yards and the width \( w = 20 \) yards.

Step 1: Calculating the Area

The formula for the area of a rectangle is \( A = l imes w \). Substitute the given values to find the area: \( A = 40 \times 20 = 800 \) square yards.

Step 2: Calculating the Perimeter

The formula for the perimeter of a rectangle is \( P = 2(l + w) \). Substitute the given values to find the perimeter: \( P = 2(40 + 20) = 2 imes 60 = 120 \) yards.

Key Concepts

Calculating Rectangle Area

The area of a rectangle refers to the total amount of space enclosed within its four sides. To calculate it, you use a very simple formula: - The formula is given by \( A = l \times w \), where \( A \) represents the area, \( l \) is the length, and \( w \) is the width.By multiplying the length by the width, you can easily find the area. For example, if the length of a rectangle is 40 yards and the width is 20 yards, you multiply these two dimensions:\[ A = 40 \times 20 = 800 \text{ square yards} \]The unit of area is always expressed in square units, like square yards, square meters, etc. Remember, the calculation gives you a measure of how much space is inside the rectangle.

Calculating Rectangle Perimeter

The perimeter of a rectangle is the total length of all its sides added together. It's like tracing the outer boundary of the rectangle. To find the perimeter, there's a straightforward formula:- The formula is \( P = 2(l + w) \), where \( P \) is the perimeter, \( l \) is the length, and \( w \) is the width.The calculation involves adding the length and the width, and then doubling that sum because a rectangle has two lengths and two widths.For instance, if the length is 40 yards and the width is 20 yards, the calculation becomes:\[ P = 2(40 + 20) = 2 \times 60 = 120 \text{ yards} \]Perimeter is expressed in linear units, such as yards, meters, etc., because it measures distance around the shape.

Geometry Formulas

Geometry often deals with shapes and their properties. Rectangles, being among the simplest shapes, have some primary formulas that help in various calculations. Understanding these formulas is essential in solving real-world and mathematical problems confidently.

• Area Formula: The formula \( A = l \times w \) calculates the amount of space inside the rectangle. It gives area in square units, indicating the size of the surface.
• Perimeter Formula: The formula \( P = 2(l + w) \) calculates the boundary length around the rectangle. Represented in linear units, it helps determine the total distance one would walk around the rectangle.

These formulas are integral to geometry and are often used in fields including engineering, architecture, and design. They not only help understand basic properties of rectangles but also form the foundation for dealing with more complex geometric problems.