91Ó°ÊÓ

Understanding the perimeter of a square is fundamental in geometry. The perimeter of a square is the total distance around the outside of the square. Since all sides of a square are equal in length, the perimeter can be easily calculated by multiplying the length of one of its sides by 4.

In our exercise, we are given the total perimeter, which is 65 feet. To find the length of one side of the square, we simply divide the perimeter by 4, using the formula:

Length of one side = Perimeter ÷ Number of sides

Which results in:

Length of one side = \( \frac{65}{4} \) feet.

When tackling geometry problems, it's often necessary to simplify mathematical expressions to arrive at an answer that's easy to understand. Simplification can involve arithmetic, such as multiplying and dividing numbers, as well as applying more complex operations like squaring numbers or working with square roots.

For our example, we need to square the side length of the square to find its area. This action gives us:

Area = \( \left(\frac{65}{4}\right)^2 \) square feet.

We then perform the squaring of \( \frac{65}{4} \), which means multiplying the fraction by itself, leading to 422.25. Thus, the simplification process helps us turn an expression into a clear, precise numeric value, which in this case is the area of the square.

Solving geometry problems often involves a step-by-step approach, where each step builds upon the previous ones. It begins with understanding the given information and determining what needs to be found. Next, we apply geometric concepts and formulas relevant to the problem.

In the exercise at hand, we first determined the side length of the square from its perimeter and then applied the formula for the area of a square, which is the square of its side. By breaking down the problem into these manageable steps, we could accurately determine that the area of the square is 422.25 square feet.

This systematic problem-solving method can be applied to a wide variety of geometry problems, empowering students to tackle even complex exercises with confidence.