Calculating the Area of a Rectangle with Given Perimeter and Length-Width Relationship

In this article, we will explore a practical problem in geometry: finding the area of a rectangle given its perimeter and the relationship between its length and width. This involves using algebraic equations and the basic formulas for the perimeter and area of a rectangle. We will break down the problem step by step to help you understand and solve similar problems efficiently.

Problem Statement

The perimeter of a rectangle is 500 meters, and its length is 50 meters more than its width. We need to find the area of the rectangle.

Step-by-Step Solution

Define Variables

Let:

W be the width of the rectangle. L be the length of the rectangle.

Given that the length is 50 meters more than the width:

L W 50

Perimeter Formula

The perimeter (P) of a rectangle is given by:

P 2L 2W

We know that the perimeter is 500 meters:

2L 2W 500

Simplifying this equation:

L W 250

Substitute the Length into the Perimeter Equation

Substitute L W 50 into the simplified perimeter equation:

W 50 W 250

Simplify:

2W 50 250

Subtract 50 from both sides:

2W 200

Divide both sides by 2:

W 100

Find the Length

Now that we have the width (W), we can find the length (L):

L W 50 100 50 150

Calculate the Area

The area (A) of the rectangle is given by:

A L × W

Substitute the values we found:

A 150 × 100 15000 m2

Therefore, the area of the rectangle is 15000 square meters.

Verification

To verify the solution:

Perimeter check: 2L 2W 2(150) 2(100) 300 200 500 meters Length and width check: L 150 meters and W 100 meters

The values satisfy the original conditions, confirming that our solution is correct.

Area Formula of a Rectangle

The formula for the area (A) of a rectangle is:

A L × W

This formula is fundamental in solving problems involving the area of rectangles. By understanding and applying this formula, you can easily find the area of any rectangle given its length and width.

Key Points to Remember

Define the variables based on the given information. Use the perimeter formula to form an equation involving the variables. Substitute the known relationships between the variables into the equation. Use algebraic operations to solve for the unknown variables. Substitute the found values into the area formula to calculate the area.

Conclusion

By following these steps, you can solve similar problems involving the area and perimeter of rectangles. Remember to always use the given relationships and formulas effectively to arrive at the correct solution.

Additional Resources

For more information on geometry and problem-solving in mathematics, consider exploring these resources:

Math is Fun Geometry Section Khan Academy Geometry Course Rectangle Area Calculator

Happy problem-solving!