Calculating the Area of a Square Given Its Perimeter

Imagine you are faced with a problem where you know the perimeter of a square but not its area. If you have a square with a perimeter of 100 cm, you can easily find its area and other properties. Let's break down the process step by step to understand and solve this problem effectively.

Understanding the Relationship Between Perimeter and Area

The perimeter of a square is the total length of its four equal sides. The formula for the perimeter (P) of a square is:

Formula: P 4s

where s is the length of one side of the square. This relationship allows us to find the side length if we know the perimeter, and vice versa.

Step-by-Step Calculation

Given: Perimeter 100 cm

Step 1: Calculate the side length of the square. Since the perimeter is 100 cm, we use the formula:

P 4s
100 4s
s 100/4
s 25 cm

Step 2: Now that we have the side length, we can find the area of the square. The area (A) of a square is given by the formula:

A s^2
A 25^2
A 625 cm^2

This means the area of the square is 625 square centimeters.

Additional Examples for Clarity

Let’s explore a few more examples to solidify your understanding of the concept:

Example 1: Perimeter 80 cm

Step 1: Calculate the side length.
P 4s
80 4s
s 80/4
s 20 cm

Step 2: Calculate the area.
A s^2
A 20^2
A 400 cm^2

Thus, the area of the square is 400 square centimeters.

Example 2: Perimeter 72 cm

Step 1: Calculate the side length.
P 4s
72 4s
s 72/4
s 18 cm

Step 2: Calculate the area.
A s^2
A 18^2
A 324 sq cm

Therefore, the area of the square is 324 square centimeters.

Understanding the Geometry

Let's reiterate the geometric principles. The perimeter of a square is the sum of all four sides. If each side is equal, then:

where a is the length of one side. The area, being the product of two equal sides, is:

Using these formulas, we can solve for the area or the side length of a square with a given perimeter.

Conclusion

Knowing the perimeter of a square allows you to calculate its area through simple mathematical operations. By understanding the formulas for perimeter (P 4s) and area (A s^2), you can easily find the area of a square. This knowledge is not just theoretical; it has practical applications in various fields, including architecture, design, and everyday problem-solving.