To find the radius of a circle, there are several methods depending on the information you have about the circle. Here are the most common approaches:

### 1. **Given the Diameter:**
The radius \( r \) is simply half of the diameter \( d \):

\[
r = \frac{d}{2}
\]

### 2. **Given the Equation of the Circle:**
If the equation of the circle is in standard form:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Here, \( (h, k) \) is the center of the circle, and \( r^2 \) is the square of the radius. To find the radius \( r \):

\[
r = \sqrt{r^2}
\]

For example, if the equation is \( (x - 2)^2 + (y + 3)^2 = 16 \), then:

\[
r = \sqrt{16} = 4
\]

### 3. **Given Two Points on a Circle:**
- **If you have the center \((h, k)\) and a point \((x_1, y_1)\) on the circle:**

  Use the distance formula to calculate the distance between these two points, which will give you the radius:

\[
r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}
\]

- **If you have two endpoints of a diameter \((x_1, y_1)\) and \((x_2, y_2)\):**

  First, find the distance between the two points (which is the diameter) using the distance formula, and then divide by 2:

\[
\text{Diameter} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
\[
r = \frac{\text{Diameter}}{2}
\]

### 4. **Given Circumference or Area:**
- **If you know the circumference \( C \) of the circle:**

  The circumference is related to the radius by the formula:

\[
C = 2\pi r
\]

  Solve for \( r \):

\[
r = \frac{C}{2\pi}
\]

- **If you know the area \( A \) of the circle:**

  The area is related to the radius by the formula:

\[
A = \pi r^2
\]

  Solve for \( r \):

\[
r = \sqrt{\frac{A}{\pi}}
\]

### Summary:
The method you use to find the radius depends on the information provided. Whether you have the diameter, the equation of the circle, two points on the circle, or even the circumference or area, you can determine the radius using appropriate formulas.

Would you like to explore one of these methods in more detail or have any other questions?

### Questions for Further Exploration:
1. How do you calculate the radius if only the circumference is known?
2. How can you derive the radius from the area of a circle?
3. What is the relationship between the radius and the diameter?
4. How does the radius influence the equation of a circle?
5. What is the importance of knowing the radius in geometric problems?
6. How can the radius be used to find the sector area of a circle?
7. How would you find the radius if given the arc length and central angle?
8. What role does the radius play in determining the properties of a circle?

### Tip:
When dealing with circles, remember that the radius is one of the most fundamental measurements. It not only defines the size of the circle but also plays a key role in many other formulas related to the circle, such as circumference, area, and even the properties of arcs and sectors.

Discover various methods to find the radius of a circle with detailed explanations and formulas. Whether given the diameter, equation, circumference, or area of a circle, learn how to calculate the radius effectively. Suitable for Grades 6-12.

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