Write the standard form of the equation of the circle with the given center and radius

Step-by-step examples of finding the center and radius of circles

Example

Find the center and radius of the circle.

???x^2+y^2+24x+10y+160=0???

In order to find the center and radius, we need to change the equation of the circle into standard form, ???(x-h)^2+(y-k)^2=r^2???, where ???h??? and ???k??? are the coordinates of the center and ???r??? is the radius.

In order to get the equation into standard form, we have to complete the square with respect to both variables.

Grouping ???x???’s and ???y???’s together and moving the constant to the right side, we get

???(x^2+24x)+(y^2+10y)=-160???

Completing the square requires us to take the coefficient on the first degree terms, divide them by ???2???, and then square the result before adding the result back to both sides.

The coefficient on the ???x??? term is ???24???, so

???\frac{24}{2}=12???

???12^2=144???

The coefficient on the ???y??? term is ???10???, so

???\frac{10}{2}=5???

???5^2=25???

Therefore, we add ???144??? inside the parentheses with the ???x??? terms, ???25??? inside the parenthesis with the ???y??? terms, and we also add ???144??? and ???25??? to the right with the ???-160???.

???(x^2+24x +144)+(y^2+10y+25)=-160 + 144+25???

Factor the parentheses and simplify the right side.

???(x+12)^2+(y+5)^2=9???

Therefore, the center of the circle is at ???(h,k)=(-12,-5)??? and its radius is ???r=\sqrt{9}=3???.

Example

What is the center and radius of the circle?

???6x^2+6y^2+12x-13=0???

In order to find the center and radius, we need to change the equation of the circle into standard form, ???(x-h)^2+(y-k)^2=r^2???, where ???h??? and ???k??? are the coordinates of the center and ???r??? is the radius.

In order to get the equation into standard form, we have to complete the square with respect to ???x???. The ???y??? term is already a perfect square.

Let’s begin by collecting like terms and moving the ???-13??? to the right.

???6x^2+12x+6y^2=13???

Our next step is to remove the coefficients of the second degree terms of the ???x??? variable and ???y??? variable. First, we’ll factor out a ???6??? then divide by ???6??? on both sides.

???6(x^2+2x+y^2)=13???

???x^2+2x+y^2=\frac{13}{6}???

Now complete the square of the ???x??? terms. The ???y??? term is already a perfect square.

???(x^2+2x)+y^2=\frac{13}{6}???

Completing the square requires us to take the coefficient on the first degree term, divide it by ???2???, then square the result before adding the result back to both sides. 

The coefficient on ???x??? is ???2???, so

???\frac{2}{2}=1???

???1^2=1???

We’ll therefore add ???1??? to both sides, and get

???(x^2+2x+1)+y^2=\frac{13}{6}+1???

Factor the ???x??? terms and simplify the right hand side.

???(x+1)^2+y^2=\frac{19}{6}???

If you want, you may also write the equation as

???(x+1)^2+(y+0)^2=\frac{19}{6}???

The center of the circle ???(h,k)??? is ???(-1,0)??? and the radius is ???\sqrt{19/6}???. Rule out ???-\sqrt{19/6}??? because a radius can't be negative.

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Created by Krishna Nelaturu

Reviewed by Aleksandra Zając, MD

Last updated: Oct 18, 2022

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:

(x−a)2+(y−b)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=−2a E=−2bF=a2+b2−r 2\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,py) lies on a circle (x-a)2 + (y-b)2 = r2, follow these steps:

  1. Substitute the coordinates of the point P(px,py) in place of x and y in LHS of the circle equation to get (px-a)2 + (py-b)2.
    • If (px-a)2 + (py-b)2 = r2, then the point P(px,py) lies on the circle.
    • If (px-a)2 + (py-b)2 ≠ r2, then the point P(px,py) does not lie on the circle.

Center coordinates and radius

Standard form: (x - A)² + (y - B)² = C

Parametric form: x = A + r cos(α), y = B + r sin(α)

General form: x² + y² + Dx + Ey + F = 0

Arc lengthArea of a circleCircle calc: find c, d, a, r… 5 more

How do you write the standard equation of a circle with the center and radius?

The equation of a circle written in the form (x−h)2+(y−k)2=r2 where (h,k) is the center and r is the radius. The circle centered at the origin with radius 1; its equation is x2+y2=1.

How do you write the equation of a circle given its radius and the center is at H K?

The equation of a circle with (h, k) center and r radius is given by:.
(x-h)2 + (y-k)2 = r2.
(x-1)2+(y-2)2 = 42.
(x2−2x+1)+(y2−4y+4) =16..
X2+y2−2x−4y-11 = 0..