This standard equation of a circle calculator is helpful to calculate the standard form of a circle equation using its center coordinates and radius, or any other form of the circle equation. With this simple tool in your hand, you can easily find the circle equation in any form you want!

In this article, we shall discuss:

  • How to write a circle equation in standard form?
  • How to convert the standard form into parametric or general form and vice versa.
  • FAQ

How to write a circle equation in standard form?

The standard equation of a circle is given by:

(xa)2+(yb)2=r2\left(x-a\right)^2 + (y-b)^2 = r^2

where:

  • (x,y)(x,y) - The coordinates of any point on the circle;
  • (a,b)(a,b) - The coordinates of the center of the circle; and
  • rr - The radius of the circle.

We can use this equation to find the standard form from its center and radius or vice versa.

How to convert the standard equation into parametric form

The parametric form of a circle equation is given by:

x=a+rcos(α)y=b+rsin(α)\begin{align*} x &= a + r\cos(\alpha)\\ y &= b + r\sin(\alpha)\\ \end{align*}

where:

  • (x,y)(x,y) - The coordinates of any point on the circle;
  • (a,b)(a,b) - The coordinates of the center of the circle;
  • rr - The radius of the circle; and
  • α\alpha - The angle subtended by the point (x,y)(x,y) at the circle's center (a,b)(a,b).

It is a straightforward conversion between these two forms; the additional parameter α\alpha is not essential for this conversion, so long as the other variables are known.

How to convert the standard equation into general form

The general form of the circle equation is an expansion of its standard form. It can be expressed as:

x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx+ Ey+ F = 0

where:

  • (x,y)(x,y) - The coordinates of any point on the circle;
  • DD - The sum of the coefficients of the x-terms;
  • EE - The sum of the coefficients of the y-terms; and
  • FF - The sum of the constant terms.

Note that the right-hand side (RHS) of this equation has to be zero. Bring every term to the left-hand side (LHS) and simplify.

As this is the expansion of the standard form, we can complete the squares of this expanded form to arrive at the standard equation and establish the following relationships between the various parameters in these two forms:

D=2aE=2bF=a2+b2r2\begin{align*} D &= -2a\\ E &= -2b\\ F &= a^2+b^2-r^2\\ \end{align*}

We can use these equations to convert between the standard form and the general form of a circle equation.

How to use this standard equation of a circle calculator

You can use this calculator for more than one thing:

  • Enter the standard equation of a circle to obtain the center, radius, and circle equation in other forms.
  • Give the center and radius of a circle to simultaneously determine its equation in all three forms.
  • Enter the equation of a circle in parametric or general form to calculate its center, radius, and equation in the standard form.

In addition, this calculator will also determine other properties of the circle, like its area and circumference.

FAQ

What is the equation of a circle with a center (0,0) and radius of 7?

x²+y² = 49. To find this equation, follow these steps:

  1. Insert the center coordinates in the place of (a,b) in the standard form of a circle equation (x-a)² + (y-b)² = r². This gives (x-0)² + (y-0)² = r².
  2. Substitute the value of radius in the place of r in this equation. This gives x²+y² = 7².
  3. Evaluate this equation to get the equation of the circle, x²+y² = 49.

How do you determine if a point lies on a circle?

To determine whether a point P(pₓ,pᵧ) lies on a circle (x-a)² + (y-b)² = r², follow these steps:

  1. Substitute the coordinates of the point P(pₓ,pᵧ) in place of x and y in LHS of the circle equation to get (pₓ-a)² + (pᵧ-b)².
    • If (pₓ-a)² + (pᵧ-b)² = r², then the point P(pₓ,pᵧ) lies on the circle.
    • If (pₓ-a)² + (pᵧ-b)² ≠ r², then the point P(pₓ,pᵧ) does not lie on the circle.
Krishna Nelaturu
Center coordinates and radius
x-coordinate
y-coordinate
Radius
Standard form: (x - A)² + (y - B)² = C
A
B
C
Parametric form: x = A + r cos(α), y = B + r sin(α)
A
B
r
General form: x² + y² + Dx + Ey + F = 0
D
E
F
Other circle properties
R
Diameter
Area
Circumference
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