How to Find the Radius of a Circle

To find the radius of a circle from its equation in general form $$ x^2 + y^2 + Dx + Ey + F = 0 $$, use this formula: $$ r = \sqrt{(-\frac{D}{2})^2 + ( -\frac{E}{2} )^2 - F } $$

If the circle's equation is given in standard form, determining the radius is much easier.

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

In this case, the center and radius of the circle are:

  • \( (h, k) \) are the coordinates of the circle's center.
  • \( r \) is the radius of the circle.

When the circle's equation is in general form:

\[ x^2 + y^2 + Dx + Ey + F = 0 \]

The center \((h, k)\) of the circle can be found using these formulas:

$$ h = -\frac{D}{2} $$

$$ k = -\frac{E}{2} $$

where \(D\) and \(E\) are the coefficients of \(x\) and \(y\) in the general circle equation \( x^2 + y^2 + Dx + Ey + F = 0 \).

Thus, the center of the circle is at coordinates \((-D/2, -E/2)\).

The radius of the circle is given by the formula:

$$ r = \sqrt{(-\frac{D}{2})^2 + ( -\frac{E}{2} )^2 - F } $$

Alternatively, to find the radius and the center of a circle, you can convert the equation from general form to standard form by completing the square and adding the necessary terms. In standard form, \( (x - h)^2 + (y - k)^2 = r^2 \), the radius is simply calculated as \( r = \sqrt{r^2} \).

A Practical Example

Example 1

In this example, we have the circle's equation in standard form:

$$ (x - 3)^2 + (y + 2)^2 = 16 $$

Finding the radius and center is straightforward.

  • The center of the circle is \( (3, -2) \).
  • The term on the right-hand side is \( r^2 = 16 \).

To find the radius \( r \), we just take the positive square root of 16:

$$ r = \sqrt{16} = 4 $$

Therefore, the radius of the circle is 4.

esempio

Example 2

In this example, the circle's equation is not in standard form:

$$ x^2 + y^2 + 6x - 8y + 9 = 0 $$

Here, we can find the radius using the radius formula:

$$ r = \sqrt{(-\frac{D}{2})^2 + ( -\frac{E}{2} )^2 - F } $$

where the coefficients in the general equation \( x^2 + y^2 + Dx + Ey + F = 0 \) are D=6, E=-8, and F=9.

$$ r = \sqrt{(-\frac{6}{2})^2 + ( -\frac{-8}{2} )^2 - 9 } $$

$$ r = \sqrt{(-3)^2 + (4)^2 - 9 } $$

$$ r = \sqrt{9 + 16 - 9 } $$

$$ r = \sqrt{16 } $$

$$ r = 4 $$

Alternatively, we can find the radius by converting the general form of the circle's equation into the standard form.

We group the \( x \) and \( y \) terms:

$$ x^2 + y^2 + 6x - 8y + 9 = 0 $$

$$ (x^2 + 6x) + (y^2 - 8y) = -9 $$

We complete the square for \( x \) by adding and subtracting the term \( (\frac{6}{2})^2 \):

$$ (x^2 + 6x +(\frac{6}{2})^2) - (\frac{6}{2})^2 + (y^2 - 8y) = -9 $$

$$ (x^2 + 6x + 3^2) - 3^2 + (y^2 - 8y) = -9 $$

$$ (x + 3)^2 - 9 + (y^2 - 8y) = -9 $$

$$ (x + 3)^2 + (y^2 - 8y) = 0 $$

We complete the square for \( y \) by adding and subtracting the term \( (\frac{8}{2})^2 \):

$$ (x + 3)^2 + (y^2 - 8y + (\frac{8}{2})^2 ) - (\frac{8}{2})^2 = 0 $$

$$ (x + 3)^2 + (y^2 - 8y + 4^2 ) - 4^2 = 0 $$

$$ (x + 3)^2 + (y - 4)^2 = 4^2 $$

$$ (x + 3)^2 + (y - 4)^2 = 16 $$

Now the equation is in standard form, and the term on the right-hand side is \( r^2 = 16 \).

To calculate the radius, we simply take the square root of this term:

$$ r = \sqrt{16} = 4 $$

Therefore, the radius of the circle is 4.

Other Useful Formulas for Finding the Radius

Generally, the formula for finding the radius \( r \) of a circle depends on the information available.

Here are some common situations:

  • If you know the circumference $$ r = \frac{C}{2\pi} $$ where \( C \) is the circumference.
  • If you know the area $$ r = \sqrt{\frac{A}{\pi}} $$ where \( A \) is the area of the circle.
  • If you know the coordinates of the endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) of the diameter, the radius is half the Euclidean distance between these points: $$ r = \frac{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}{2} $$

These formulas allow you to calculate the radius of a circle in various situations. They are included for completeness.

And so on.

 
 

Please feel free to point out any errors or typos, or share suggestions to improve these notes. English isn't my first language, so if you notice any mistakes, let me know, and I'll be sure to fix them.

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