Are you looking for a general to standard form of a circle calculator for mathematics? Then you have come to the right place. Our general to standard form of a circle calculator will help you convert from general to standard form with ease. Best of all, you do not need to know any formulas to use it. However, if you would like to understand how we went about converting from general to standard form or would like to do some conversions on your own, we have all the information you will need to get the job done. Continue reading to learn:

  • What the equation of a circle means;
  • What is the equation for the general form of a circle;
  • What is the equation for the standard form of a circle; and
  • The process used to convert from general to the standard form.

How to use our general to standard form of a circle calculator:

Our general to standard form of a circle calculator allows you to change from standard to general form by entering the values in the section marked standard form. Once you have don't this, the values for the general form of the circle will appear in real-time.

What do we mean by equation of a circle?

Have you been wondering what exactly we mean when we say the equation of a circle?

The equation of a circle is the formula we use to show the circle's position in the cartesian plane. The equation represents the points along the circumference of the circle.

There are several formulas that we may use to find the equation of the circle. However, this calculator will only look at the equations for the standard and general forms.

What is the general form equation of a circle?

The general form of the equation of a circle is x² + y² + Dx + Ey + F.

In this equation, D, E, and F are real numbers. To find the center and radius from the general form, we need to convert this equation to its standard form.

What is the equation of a circle in standard form?

The standard form of the equation of a circle is (x - h)² + (y - k)² = r².

This is the equation of a circle in standard form with center h, k, and radius r. Because this equation gives the coordinates and radius, it is excellent for drawing the circle in the Cartesian plane.

How to convert from general to standard form of a circle?

Here is what you do to convert from the general to the standard form of a circle:

  • Enter the values for D, E, and F into our general to standard form of a circle calculator;
  • The values in the standard form section are for the radius and the center coordinates.
  • Lastly, substitute the new values into the equations shown above the calculator.

Not satisfied with this? Are you trying to understand how to convert from the general to the standard form of a circle for school and need to learn the entire process? The following explanation is just what you need:

Let's take the equation for the general form of a circle:

x2+y2+Dx+Ey+F=0\text{x}^2 + \text{y}^2 + \text{Dx} + \text{Ey} + \text{F} = 0

Before we proceed, let us replace DD, EE, and FF with numbers.

x2+y2+22x12y8=0\text{x}^2 + \text{y}^2 + 22\text{x} - 12\text{y} - 8 = 0

So the first thing you need to do with this equation is, move the constant term to the other side of the equation. To move 8 to the other side, we need to add 8 to both sides. Having done that, you should have the following result:

x2+y2+22x12y=8\text{x}^2 + \text{y}^2 + 22\text{x} - 12\text{y} = 8

Next, group all the xx terms and all the yy terms. The resulting equation will be:

(x2+22x)+(y212y)=8(\text{x}^2 + 22\text{x}) + (\text{y}^2 - 12\text{y}) = 8

Completing the square

Now, let's complete the square. We do this by halving the coefficient of xx and yy, then squaring them. Since the coefficient of xx is 22, half of 22 would be 11. When we square 11, we get 121. So the first bracket should now look like this:

(x2+22x+121)(\text{x}^2 + 22\text{x} + 121)

Since the coefficient of y is 12 and half of 12 is 6. When you square 6, you will get 36. The second bracket will be:

(y212y+36)(\text{y}^2 - 12\text{y} + 36)

An equation must always be balanced. So, because we added 121 and 36 to one side of the equation, we need to also add those numbers to the other side of the equation.

Our resulting equation is:
(x2+22x+121)+(y212y+36)=165(\text{x}^2 + 22\text{x} + 121) + (\text{y}^2 - 12\text{y} + 36) = 165

Factorization

Next, you need to factorize this equation:

To do this, we go to the first bracket and take the square root of the first term, take the sign of the middle term, and take the square root of the last term. We repeat this process with the second bracket as well.

Your resulting equation should be:
(x+11)2+(y6)2=165(\text{x} + 11)^2 + (\text{y} - 6)^2 = 165

This final equation is the standard form of the circle.

Adena Benn
x² + y² + Dx + Ey + F = 0
D
E
F
(x - A)² + (y - B)² = C
A
B
C
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