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 = \frac{2

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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 is essential to understand the formula and its equivalent equations. The formula for the circumference of a circle is given by C=2πrC = 2 \pi r, where CC is the circumference and rr is the radius of the circle. In this article, we will explore the equivalent equations of the circumference formula and provide a detailed explanation of each option.

The formula C=2πrC = 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 twice the product of the radius and the mathematical constant pi (π\pi). The radius of a circle is the distance from the center of the circle to the edge, and it is denoted by the symbol rr.

Now that we have understood the formula for the circumference of a circle, let's explore the equivalent equations. The equivalent equations are the equations that can be derived from the original formula by rearranging the terms. In other words, the equivalent equations are the equations that have the same solution as the original formula.

Option A: r=Cπr = \frac{C}{\pi}

The first option is r=Cπr = \frac{C}{\pi}. To derive this equation, we can start with the original formula C=2πrC = 2 \pi r and divide both sides by 2π2 \pi. This gives us r=C2πr = \frac{C}{2 \pi}. However, this is not the correct option. To get the correct option, we can multiply both sides of the equation by 12\frac{1}{2}, which gives us r=C2π×12=C2π×22=Cπr = \frac{C}{2 \pi} \times \frac{1}{2} = \frac{C}{2 \pi} \times \frac{2}{2} = \frac{C}{\pi}.

Option B: r=C(2π)r = C(2 \pi)

The second option is r=C(2π)r = C(2 \pi). This option is incorrect because it is not an equivalent equation of the original formula. The correct equivalent equation is r=C2πr = \frac{C}{2 \pi}, not r=C(2π)r = C(2 \pi).

Option C: r=C2πr = \frac{C}{2 \pi}

The third option is r=C2πr = \frac{C}{2 \pi}. This option is correct because it is an equivalent equation of the original formula. To derive this equation, we can start with the original formula C=2πrC = 2 \pi r and divide both sides by 2π2 \pi. This gives us r=C2πr = \frac{C}{2 \pi}.

Option D: r=2Cr = \frac{2}{C}

The fourth option is r=2Cr = \frac{2}{C}. This option is incorrect because it is not an equivalent equation of the original formula. The correct equivalent equation is r=C2πr = \frac{C}{2 \pi}, not r=2Cr = \frac{2}{C}.

In conclusion, the equivalent equation of the circumference formula C=2πrC = 2 \pi r is r=C2πr = \frac{C}{2 \pi}. This equation can be derived by dividing both sides of the original formula by 2π2 \pi. The other options are incorrect because they are not equivalent equations of the original formula.

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

The circumference of a circle is a fundamental concept in mathematics, and it is essential to understand the formula and its equivalent equations. In our previous article, we explored the equivalent equations of the circumference formula and provided a detailed explanation of each option. In this article, we will answer some of the most frequently asked questions about the circumference of a circle.

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

A: The formula for the circumference of a circle is given by C=2πrC = 2 \pi r, where CC is the circumference and rr is 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 the edge. It is denoted by the symbol rr.

Q: What is the circumference of a circle with a radius of 4 cm?

A: To find the circumference of a circle with a radius of 4 cm, we can use the formula C=2πrC = 2 \pi r. Plugging in the value of r=4r = 4 cm, we get C=2π(4)=8πC = 2 \pi (4) = 8 \pi cm.

Q: What is the equivalent equation of the circumference formula?

A: The equivalent equation of the circumference formula is r=C2πr = \frac{C}{2 \pi}.

Q: How do I derive the equivalent equation of the circumference formula?

A: To derive the equivalent equation of the circumference formula, we can start with the original formula C=2πrC = 2 \pi r and divide both sides by 2π2 \pi. This gives us r=C2πr = \frac{C}{2 \pi}.

Q: What is the difference between the circumference and the diameter of a circle?

A: The circumference of a circle is the distance around the circle, while the diameter is the distance across the circle, passing through its center. The circumference is always longer than the diameter.

Q: How do I calculate the circumference of a circle with a diameter of 10 cm?

A: To find the circumference of a circle with a diameter of 10 cm, we can use the formula C=πdC = \pi d. Plugging in the value of d=10d = 10 cm, we get C=π(10)=10πC = \pi (10) = 10 \pi cm.

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

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

Q: How do I use the circumference formula to find the radius of a circle?

A: To find the radius of a circle using the circumference formula, we can rearrange the formula to solve for rr. This gives us r=C2πr = \frac{C}{2 \pi}.

In conclusion, the circumference of a circle is a fundamental concept in mathematics, and it is essential to understand the formula and its equivalent equations. We hope that this Q&A guide has provided you with a better understanding of the circumference of a circle and its applications.

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