The Circumference Of A Circle Can Be Found Using The Formula $C = 2 \pi R$.Which Is An Equivalent Equation Solved For $r$?A. $r = \frac{C}{\pi}$ B. $r = C(2 \pi$\] C. $r = \frac{C}{2 \pi}$ D. $r =

The Circumference of a Circle: Understanding the Formula and Its Equivalent Equations

The circumference of a circle is a fundamental concept in mathematics, and it plays a crucial role in various mathematical and real-world applications. The formula for the circumference of a circle is given by $C = 2 \pi r$, where $C$ represents the circumference and $r$ represents the radius of the circle. In this article, we will explore the equivalent equation solved for $r$, which is a crucial concept in understanding the relationship between the circumference and the radius of a circle.

The formula $C = 2 \pi r$ is a fundamental concept in mathematics, and it is used to calculate the circumference of a circle. The formula states that the circumference of a circle is equal to $2$ times the product of the radius and the mathematical constant $\pi$. The radius of a circle is the distance from the center of the circle to any point on the circumference.

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To solve for $r$, we need to isolate the variable $r$ in the formula $C = 2 \pi r$. This can be done by dividing both sides of the equation by $2 \pi$. The resulting equation is:

$$r = \frac{C}{2 \pi}$$

This equation states that the radius of a circle is equal to the circumference divided by $2 \pi$.

Now, let's examine the equivalent equations given in the options:

• Option A: $r = \frac{C}{\pi}$
• Option B: $r = C(2 \pi)$
• Option C: $r = \frac{C}{2 \pi}$
• Option D: $r = C \pi$

Analyzing Option A

Option A states that $r = \frac{C}{\pi}$. However, this equation is not equivalent to the original formula $C = 2 \pi r$. To see why, let's substitute the value of $C$ from the original formula into Option A:

$$r = \frac{2 \pi r}{\pi}$$

Simplifying the equation, we get:

$$r = 2r$$

This equation is not true, as the radius of a circle cannot be equal to twice its value. Therefore, Option A is not an equivalent equation.

Analyzing Option B

Option B states that $r = C(2 \pi)$. However, this equation is not equivalent to the original formula $C = 2 \pi r$. To see why, let's substitute the value of $C$ from the original formula into Option B:

$$r = C(2 \pi)$$

$$r = 2 \pi r(2 \pi)$$

Simplifying the equation, we get:

$$r = 4 \pi^2 r$$

This equation is not true, as the radius of a circle cannot be equal to four times its value multiplied by the square of $\pi$. Therefore, Option B is not an equivalent equation.

Analyzing Option C

Option C states that $r = \frac{C}{2 \pi}$. This equation is equivalent to the original formula $C = 2 \pi r$. To see why, let's substitute the value of $C$ from the original formula into Option C:

$$r = \frac{2 \pi r}{2 \pi}$$

Simplifying the equation, we get:

$$r = r$$

This equation is true, as the radius of a circle is indeed equal to itself. Therefore, Option C is an equivalent equation.

Analyzing Option D

Option D states that $r = C \pi$. However, this equation is not equivalent to the original formula $C = 2 \pi r$. To see why, let's substitute the value of $C$ from the original formula into Option D:

$$r = C \pi$$

$$r = 2 \pi r \pi$$

Simplifying the equation, we get:

$$r = 2 \pi^2 r$$

This equation is not true, as the radius of a circle cannot be equal to two times its value multiplied by the square of $\pi$. Therefore, Option D is not an equivalent equation.

In conclusion, the equivalent equation solved for $r$ is $r = \frac{C}{2 \pi}$. This equation is derived from the original formula $C = 2 \pi r$ by dividing both sides of the equation by $2 \pi$. The other options, A, B, and D, are not equivalent equations, as they do not satisfy the original formula. Therefore, the correct answer is Option C, $r = \frac{C}{2 \pi}$.

The final answer is: $\boxed{C}$
The Circumference of a Circle: A Q&A Guide

In our previous article, we explored the formula for the circumference of a circle and its equivalent equations. In this article, we will answer some frequently asked questions about the circumference of a circle and provide additional insights into this fundamental concept in mathematics.

Q: What is the formula for the circumference of a circle?

A: The formula for the circumference of a circle is given by $C = 2 \pi r$, where $C$ represents the circumference and $r$ represents the radius of the circle.

Q: What is the radius of a circle?

A: The radius of a circle is the distance from the center of the circle to any point on the circumference.

Q: How do I calculate the circumference of a circle?

A: To calculate the circumference of a circle, you need to know the radius of the circle. You can then use the formula $C = 2 \pi r$ to calculate the circumference.

Q: What is the equivalent equation solved for $r$?

A: The equivalent equation solved for $r$ is $r = \frac{C}{2 \pi}$.

Q: Why is the formula $C = 2 \pi r$ important?

A: The formula $C = 2 \pi r$ is important because it allows us to calculate the circumference of a circle given its radius. This formula has numerous applications in mathematics, science, and engineering.

Q: Can I use the formula $C = 2 \pi r$ to calculate the area of a circle?

A: No, the formula $C = 2 \pi r$ is used to calculate the circumference of a circle, not its area. To calculate the area of a circle, you need to use the formula $A = \pi r^2$.

Q: What is the relationship between the circumference and the radius of a circle?

A: The circumference of a circle is directly proportional to its radius. As the radius of a circle increases, its circumference also increases.

Q: Can I use the formula $C = 2 \pi r$ to calculate the circumference of a circle with a diameter of 10 cm?

A: Yes, you can use the formula $C = 2 \pi r$ to calculate the circumference of a circle with a diameter of 10 cm. First, you need to find the radius of the circle, which is half of the diameter. The radius is 5 cm. Then, you can use the formula $C = 2 \pi r$ to calculate the circumference:

$$C = 2 \pi (5)$$

$$C = 10 \pi$$

Therefore, the circumference of the circle is approximately 31.4 cm.

In conclusion, the circumference of a circle is a fundamental concept in mathematics that has numerous applications in science and engineering. The formula $C = 2 \pi r$ allows us to calculate the circumference of a circle given its radius. We hope that this Q&A guide has provided you with a better understanding of the circumference of a circle and its equivalent equations.

The final answer is: $\boxed{C = 2 \pi r}$