Problem 6 - Algebra for College Students

Question

A circle has area \(25 \pi \mathrm{ft}^{2}\). What would be the perimeter of a square whose sides each measure the same length as the radius of the circle?

Step-by-Step Solution

Answer

The perimeter of the square is 20 feet.

Step 1: Find the radius of the circle

Use the formula for the area of a circle, which is \( A = \pi r^2 \). Given the area is \( 25 \pi \mathrm{ft}^2\), solve for \( r \) by setting \( \pi r^2 = 25 \pi \). Divide both sides by \pi\: \( r^2 = 25\). Then take the square root of both sides to find \( r \): \( r = 5 \mathrm{ft} \).

Step 2: Relate the radius to the side of the square

Since the radius of the circle is the same as the side length of the square, each side of the square measures 5 feet.

Step 3: Calculate the perimeter of the square

The perimeter of a square is found by adding all four sides. Since each side is 5 feet, the perimeter \( P \) can be calculated as: \ P = 4 \times 5 \mathrm{ft}\ = 20 \mathrm{ft} \.

Key Concepts

circle area

Understanding the area of a circle is crucial for solving many geometry problems. The formula for the area of a circle is: \( A = \pi r^2 \), where \( A \) stands for the area, and \( r \) represents the radius of the circle. In this exercise, we are given the area as \( 25 \pi \) square feet. Our goal is to find the radius. To do this, we equate \( \pi r^2 \) to \( 25 \pi \). Dividing both sides by \pi \, we get \( r^2 = 25 \). Taking the square root of both sides, the radius \( r \) is found to be 5 feet.

square perimeter

Once we know the radius of the circle, the next step is to work with the square. The perimeter of a square is the total length around it and is calculated by adding the length of all its sides. The formula for the perimeter of a square is: \( P = 4s \), where \( s \) is the side length. In our problem, the side of the square is the same as the radius of the circle, which is 5 feet. Therefore, the perimeter is: \ P = 4 \times 5 \text{ ft} = 20 \text{ ft} \.

radius to side length conversion

In geometry, converting various measurements is a common task. Here, we convert the radius of a circle to the side length of a square. Both the radius of the circle and the side of the square describe linear distances. Given that the radius is 5 feet, and knowing that one side of our square is equivalent to this radius, the side length of the square is also 5 feet. This direct conversion is essential for our calculation of the square's perimeter.

geometry problem solving

Solving geometry problems involves understanding and applying several concepts. For the exercise at hand:

• First, recognize the need to find the radius from the area of the circle using the formula \( A = \pi r^2 \).
• Next, translate the radius to the side length of the square, knowing they are equivalent in this case.
• Finally, apply the formula for the perimeter of the square \( P = 4s \).

By breaking down the problem into manageable parts, you simplify the complex tasks. Always relate concepts back to the basic formulas to ensure accuracy in solving geometry problems.