In The Standard \[$(x, Y)\$\] Coordinate Plane, The Graph Of \[$(x+3)^2+(y+5)^2=16\$\] Is A Circle. What Is The Circumference Of The Circle, Expressed In Coordinate Units?A. \[$4 \pi\$\] B. \[$5 \pi\$\] C. \[$3

In the standard $(x, y)$ coordinate plane, the equation of a circle can be expressed in the form $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ represents the coordinates of the center of the circle, and $r$ represents the radius of the circle. The given equation $(x+3)^2 + (y+5)^2 = 16$ can be rewritten as $(x-(-3))^2 + (y-(-5))^2 = 4^2$, which indicates that the center of the circle is at the point $(-3, -5)$ and the radius of the circle is $4$ units.

Calculating the Circumference of a Circle

The circumference of a circle is given by the formula $C = 2\pi r$, where $r$ represents the radius of the circle. In this case, the radius of the circle is $4$ units, so the circumference of the circle can be calculated as follows:

$C = 2\pi r$ $C = 2\pi(4)$ $C = 8\pi$

Therefore, the circumference of the circle is $8\pi$ units.

Comparing the Answer Choices

The answer choices provided are:

A. $4\pi$ B. $5\pi$ C. $3\pi$ D. $8\pi$

Based on the calculation above, the correct answer is:

D. $8\pi$

Conclusion

In this problem, we were given the equation of a circle in the standard $(x, y)$ coordinate plane and asked to find the circumference of the circle. We first identified the center and radius of the circle from the equation, and then used the formula for the circumference of a circle to calculate the answer. The correct answer is $8\pi$ units.

Key Takeaways

• The equation of a circle in the standard $(x, y)$ coordinate plane is given by $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ represents the coordinates of the center of the circle, and $r$ represents the radius of the circle.
• The circumference of a circle is given by the formula $C = 2\pi r$, where $r$ represents the radius of the circle.
• To find the circumference of a circle, we need to identify the radius of the circle from the equation and then use the formula for the circumference of a circle.

Additional Practice Problems

If you want to practice more problems like this, here are a few additional problems:

• Find the circumference of a circle with a radius of $6$ units.
• Find the circumference of a circle with a radius of $2$ units.
• Find the circumference of a circle with a radius of $10$ units.

You can try to solve these problems on your own, and then check your answers with the solutions provided below.

Solutions to Additional Practice Problems

• Find the circumference of a circle with a radius of $6$ units.

$C = 2\pi r$ $C = 2\pi(6)$ $C = 12\pi$

• Find the circumference of a circle with a radius of $2$ units.

$C = 2\pi r$ $C = 2\pi(2)$ $C = 4\pi$

• Find the circumference of a circle with a radius of $10$ units.

$$ $$ $$

In the previous article, we discussed how to find the circumference of a circle given its equation in the standard $(x, y)$ coordinate plane. In this article, we will answer some frequently asked questions about circles and their circumference.

Q: What is the equation of a circle?

A: The equation of a circle in the standard $(x, y)$ coordinate plane is given by $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ represents the coordinates of the center of the circle, and $r$ represents the radius of the circle.

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

A: To find the circumference of a circle, you need to identify the radius of the circle from the equation and then use the formula $C = 2\pi r$, where $r$ represents the radius of the circle.

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

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

Q: How do I find the radius of a circle from its equation?

A: To find the radius of a circle from its equation, you need to rewrite the equation in the standard form $(x-h)^2 + (y-k)^2 = r^2$. The value of $r$ in this equation is the radius of the circle.

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

Q: How do I find the diameter of a circle?

A: To find the diameter of a circle, you need to multiply the radius of the circle by $2$. The formula for the diameter of a circle is $d = 2r$, where $r$ represents the radius of the circle.

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

A: The circumference of a circle is always equal to $\pi$ times the diameter of the circle. The formula for this relationship is $C = \pi d$, where $C$ represents the circumference of the circle, and $d$ represents the diameter of the circle.

Q: Can I find the circumference of a circle if I know its diameter?

A: Yes, you can find the circumference of a circle if you know its diameter. You can use the formula $C = \pi d$, where $C$ represents the circumference of the circle, and $d$ represents the diameter of the circle.

Q: What is the value of $\pi$?

A: The value of $\pi$ is approximately $3.14159$. However, for most mathematical calculations, you can use the approximation $\pi \approx 3.14$.

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

A: To use a calculator to find the circumference of a circle, you need to enter the value of the radius of the circle and then multiply it by $2\pi$. For example, if the radius of the circle is $4$, you can enter the value $4$ and then multiply it by $2\pi$ to get the circumference.

Conclusion

In this article, we answered some frequently asked questions about circles and their circumference. We discussed how to find the circumference of a circle given its equation, how to find the radius of a circle from its equation, and how to use the formula for the circumference of a circle. We also discussed the relationship between the circumference and the diameter of a circle, and how to use a calculator to find the circumference of a circle.