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General to Standard Form Circle Converter

The calculator takes the equation of a circle in general form, with variables for x, y, and constants a, b, c, d and e, and converts it to the standard form equation for a circle with variables h, k, and r. It then calculates the center of the circle (h, k) and its radius r. If the equation cannot be converted to standard form, the calculator reports an error message.

This page exists due to the efforts of the following people:

Timur

Timur

Created: 2021-10-05 13:04:06, Last updated: 2023-04-02 15:56:28

PLANETCALC, Center and radius of a circle by going from general form to standard form

Center and radius of a circle by going from general form to standard form

Enter a, b, c, d, e coefs of general equation of a circle in the following order: ax² + by² + cx + dy + e = 0
Digits after the decimal point: 2
General form equation of a circle
 
Standard form equation of the circle
 

Validation

Radius of the circle
 
Center of the circle
 

General form to standard form calculation

The calculator above can be used for problems on an equation of a circle in a general form. Most often you use an equation of a circle in a standard form, that is
(x-a)^2+(y-b)^2=R^2
From this form of a circle equation, you can easily pick the center of a circle - this would be a point with (a,b) coordinates, and the radius of a circle - this would be a square root of a right part of the equation.

However, if we square the brackets and move the right part of the equation to the left, it will look something like that:
x^2+y^2+cx+dy+e=0
And this is the general form of the equation of a circle. Here you can't easily tell the center and the radius, and most of the problems ask you to find the center and the radius exactly from this form.

The solution is to convert the general form back to the standard form. Regroup the members like so:
(x^2+cx) + (y^2+dy)+e=0

Now you can use the technique know as completing the square (for more details see Completing the square). This means that you should replace the expression like ax^2+bx+c with the expression like a(x-h)^2+k. Note that for the general form of a circle's equation the first two coefficients are ones, and the free member could be assigned zero. The formulas for h and k become much simplier.

For x^2+cx:
h_x=-\frac{c}{2}\\k_x=-\frac{c^2}{4}

For y^2+dy:
h_y=-\frac{d}{2}\\k_y=-\frac{d^2}{4}

Then
(x^2+cx) + (y^2+dy)+e=0 \\ \to (x-h_x)^2+k_x + (y-h_y)^2+k_y + e=0 \\ \to (x-h_x)^2 + (y-h_y)^2=-e - k_x - k_y

In the end, we will get our equation of the circle back to the standard form, and picking the center and the radius becomes trivial. Note the caveat: If you get the negative number on the right, then it was NOT a general form of an equation of a circle at all (sometimes they ask you to check this). The calculator above also validates this condition.

For the opposite problem - going from the center and the radius of a circle to the general form or from standard form (which is easily constructed from the center and the radius) to the general form, you can use the calculator Equation of a circle calculator

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 #circle #geometry Algebra circle general form Geometry radius standard form
PLANETCALC, General to Standard Form Circle Converter
Timur2023-04-02 15:56:28

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