Problem 91 - Algebra for College Students

Question

Set up an equation and solve each of the following problems. Suppose that the area of a circle is numerically equal to the perimeter of a square and that the length of a radius of the circle is equal to the length of a side of the square. Find the length of a side of the square. Express your answer in terms of \(\pi\).

Step-by-Step Solution

Answer

The side of the square is \( \frac{4}{\pi} \).

Step 1: Understanding the Problem

We need to set up equations for the area of a circle and the perimeter of a square, given that these are equal. Additionally, the radius of the circle and the side of the square are equal.

Step 2: Identify Formulas

Recall the formulas: The area of a circle is given by the formula \( A = \pi r^2 \), and the perimeter of a square is given by the formula \( P = 4s \). Here, \( r \) is the radius of the circle, and \( s \) is the side length of the square.

Step 3: Set Up the Equation

We know that the area of the circle is equal to the perimeter of the square, so we write the equation \( \pi r^2 = 4s \). Additionally, since the radius is equal to the side of the square, we have \( r = s \).

Step 4: Substitute and Simplify

Substitute \( s \) for \( r \) in the equation \( \pi r^2 = 4s \). This gives \( \pi s^2 = 4s \).

Step 5: Solve the Equation

Divide both sides of the equation \( \pi s^2 = 4s \) by \( s \) (assuming \( s eq 0 \)). This simplifies to \( \pi s = 4 \). Solve for \( s \) by dividing both sides by \( \pi \), resulting in \( s = \frac{4}{\pi} \).

Step 6: Conclusion and Verification

The solution shows that \( s = \frac{4}{\pi} \) satisfies the condition that both the area of the circle and the perimeter of the square are the same and the side equal to the radius.

Key Concepts

Circle Area Formula

The area of a circle is a key concept in geometry. To find the area, you use the formula \( A = \pi r^2 \). This tells you that the area is equal to pi times the radius squared.
Pi, represented as \( \pi \), is approximately 3.14159 and is a mathematical constant that represents the ratio of a circle's circumference to its diameter.

• \( r \) is the radius, or the distance from the center of the circle to any point on its edge.
• To calculate the area, you first square the radius and then multiply the result by \( \pi \).

Understanding how to calculate the area of a circle is crucial in many math problems and real-world applications such as determining the size of circular plots or materials needed for circular objects.

Square Perimeter Formula

The perimeter of a square is much simpler to calculate than a circle's area. The perimeter is the total length around the square, and it is calculated using the formula \( P = 4s \).
Here:

• \( s \) represents the length of each side of the square.
• Since all sides of a square are of equal length, you simply multiply the side length by four.

This formula helps you quickly find out how much material you might need to enclose a square area, such as fencing or bordering.

Equation Setup

Setting up an equation involves expressing a problem in terms of mathematical expressions and finding relationships between different quantities. In this exercise, we relate the area of a circle to the perimeter of a square.

• We know from our formulas that \( A = \pi r^2 \) for the circle and \( P = 4s \) for the square.
• The problem gives that these values are equal, leading to the equation \( \pi r^2 = 4s \).
• Additionally, since the radius of the circle is equal to the side of the square, we say \( r = s \).

This translation of words into equations is a foundational part of solving algebraic equations.

Substitution Method

The substitution method is a simple and effective technique to solve equations, especially when dealing with two variables that are related.
By substituting one variable with another, we simplify the equation. In our problem, since \( r = s \), we replace \( r \) in the circle's area formula with \( s \).

• The equation \( \pi r^2 = 4s \) becomes \( \pi s^2 = 4s \).
• We then divide through by \( s \) (assuming \( s eq 0 \)) to get \( \pi s = 4 \), and solve for \( s \).

Using the substitution method here allows us to efficiently find that the side of the square, \( s \), is \( \frac{4}{\pi} \). This solution shows the relationship between the geometrical properties given in the problem.