Solving the Problem: Equal Perimeters of a Rectangle and a Square

In geometry, understanding the relationship between different shapes is a fascinating and critical skill. This article will guide you through a problem where a rectangle and a square share the same perimeter. Specifically, we will explore a scenario where a rectangle has a length of 8 cm and an area of 32 cm2. The goal is to find the length of each side of the square that has the same perimeter as the rectangle.

Understanding the Problem

The problem can be broken down into a few clear and logical steps. First, we need to find the width of the rectangle. We know that the area of a rectangle is given by the formula Length x Width. Given that the area of the rectangle is 32 cm2 and the length is 8 cm, we can easily calculate the width.

Mathematically, this can be shown as:

A L x W 32 8 x W W 32 / 8 W 4 cm

So, the width of the rectangle is 4 cm. The perimeter of a rectangle is calculated as 2(L W). Substituting the known values:

P 2(8 4) P 2 x 12 P 24 cm

This means the perimeter of the rectangle is 24 cm. Since the square has the same perimeter, we can use this perimeter to find the length of each side of the square.

Find the Side Length of the Square

The perimeter of a square is 4 times the length of one of its sides. If we let 's' represent the side length of the square, we can express the perimeter of the square as:

P 4s

We know the perimeter of the square is 24 cm, so we can set up the equation:

4s 24 s 24 / 4 s 6 cm

This means the length of each side of the square is 6 cm.

Key Concepts and Mathematical Equations

The key concepts in this problem include:

The relationship between the area and the dimensions of a rectangle (Length x Width Area) The formula for the perimeter of a rectangle (2(Length Width)) The formula for the perimeter of a square (4 x Side Length)

The problem also demonstrates the importance of understanding and applying mathematical equations to solve geometric problems. By breaking down the problem into smaller, manageable steps, we can find the solution efficiently.

Conclusion

In conclusion, a rectangle with a length of 8 cm and an area of 32 cm2 has a width of 4 cm. The perimeter of this rectangle is 24 cm, which is also the perimeter of a square with each side measuring 6 cm. Understanding the relationship between the dimensions and perimeters of these shapes is crucial in solving similar geometric problems.