The Equation of a Circle Calculator is a free online tool that displays the equation of a circle for any input. With Studyqueries’s online equation of a circle calculator, the calculation is faster, and the equation is displayed in a fraction of a second.

How to Use the Equation of a Circle Calculator?

Here is how to use the equation of a circle calculator:

  • Step 1: Type the circle’s radius and center in the corresponding fields
  • Step 2: Now click to get the circle’s equation by clicking the “Find Equation of Circle” button
  • Step 3: In the new window, the equation for a circle corresponding to the given input will be displayed

Equation Of A Circle Calculator


Circles have the following standard equation:

(x-h)² + (y-k)² = r²

r is the radius of the circle, and (h,k) is the coordinates of the circle’s center.

Before we deduce the equation for a circle, let us discuss what is a circle? A circle is a set of all points that are equally spaced from a fixed point in a plane. Known as the center of the circle, this fixed point is the fixed point of the circle. Circle radius refers to the distance between a point on the circumference and the center. As we shall see in this article, the standard form of an equation of a circle and examples of an equation of a circle whose center occurs at the origin and one whose center does not occur at the origin are discussed.

What is the Equation of a Circle?

The circumference is defined as the distance between all points on a curve from the fixed point, called the center, and all points on that curve. A circle with a center of (h,k) and a radius of r has the following equation:

(x-h)² + (y-k)² = r²

If we know the coordinates of the circle’s center and its radius, we can easily find its equation.

As an example: Let’s say point (1,2) is the center of the circle, and the radius is equal to 4 cm. Therefore, the equation of this circle will be:

(x-1)²+(y-2)² = 4²

(x²−2x+1)+(y²−4y+4) =16

x²+y²−2x−4y-11 = 0

Equation Of Circle Is Function or Not?

The question of whether a circle can be considered a function arises in the case of circles. Circles are not functions, as it should be clear. Since a function is defined by its values associated with one point in the codomain, while the line that passes through the circle intersects it at two points.

Circles are described mathematically by equations. Circle equations are presented here in all their forms, such as standard and general forms, with examples.

Equation of a Circle When the Centre is Origin

Imagine an arbitrary point P(x, y) on the circle. Let ‘a’ be the radius of the circle that is equal to OP.

Centre of Circle is Origin
Centre of Circle is Origin

We know that the distance between the point (x, y) and the origin (0,0)can be calculated using the distance formula, which is equal to

√[x²+ y²]= a

A circle whose center serves as its origin has the equation:

x²+y²= a²

Where “a” is the radius of the circle.

Alternative Method

We can also derive in another way. Imagine that (x,y) is a point on a circle, and that the center of the circle is (0,0). The radius of the circle is the hypotenuse of the right triangle formed by the perpendicular line drawn from point (x) to the x-axis. The base of the triangle is the distance along the x-axis, while the height is the distance along the y-axis. We can apply Pythagoras’ theorem to the problem as follows:

x²+y² = (radius)²

Equation of a Circle When the Centre is not an Origin

C(h, k) represents the center of the circle, and P(x, y) represents any point on the circle.

Therefore, the radius of a circle is CP.

Centre of Circle is not Origin
Centre of Circle is not Origin

By using distance formula,

(x-h)² + (y-k)² = (CP)²

Let radius be ‘a’.

In this case, the equation of the circle with center (h, k)and radius ‘a’ is,

(x-h)²+(y-k)² = a²

which is called the standard form for the equation of a circle.

Equation of a Circle in General Form

The general equation of any type of circle is represented by:

x² + y² + 2gx + 2fy + c = 0, for all values of g, f and c.

Adding g2 + f2 on both sides of the equation gives,

x² + 2gx + g²+ y² + 2fy + f²= g² + f² − c ………………(1)

Since, (x+g)² = x²+ 2gx + g² and (y+f)² =y² + 2fy + f² substituting the values in equation (1), we have

(x+g)²+ (y+f)² = g² + f²−c …………….(2)

Comparing (2) with (x−h)² + (y−k)² = a², where (h, k) is the center and ‘a’ is the radius of the circle.

h=−g, k=−f

a² = g²+ f²−c

Therefore,

x² + y² + 2gx + 2fy + c = 0, represents the circle with centre (−g,−f) and radius equal to a² = g² + f²− c.

If g² + f² > c, then the radius of the circle is real.

If g² + f² = c, then the radius of the circle is zero which tells us that the circle is a point that coincides with the center. Such a type of circle is called a point circle

g² + f² <c, then the radius of the circle becomes imaginary. Therefore, it is a circle having a real center and imaginary radius.

Other Circle Formulas

The following formulas are given for circles in terms of radius.

Diameter= 2 x radius

Circumference= 2π (radius)

Area= π(radius)2

How to Find the Equation of the Circle?

The following are some solved problems for finding the equation of a circle in both cases, such as when the circle’s center is an origin and when the center is not an origin.

Example 1: Consider a circle whose center is at the origin and whose radius is 8.

Solution:

Given: Centre is (0, 0), radius is 8 units.

We know that the equation of a circle when the center is origin:

x²+ y² = a²

For the given condition, the equation of a circle is given as

x² + y² = 8²

x² + y²= 64, which is the equation of a circle

Example 2: Find the equation of the circle whose center is (3,5) and whose radius is 4.

Solution:

Here, the center of the circle is not an origin.

Therefore, the general equation of the circle is,

(x-3)² + (y-5)² = 4²

x² – 6x + 9 + y² -10y +25 = 16

x² +y² -6x -10y + 18 =0

Example 3: The equation of a circle is x²+y²−12x−16y+19=0. Find the center and radius of the circle.

Solution:

Given equation is of the form x²+ y² + 2gx + 2fy + c = 0,

2g = −12, 2f = −16,c = 19

g = −6,f = −8

Centre of the circle is (6,8)

Radius of the circle = √[(−6)² + (−8)² − 19 ]= √[100 − 19] =

= √81 = 9 units.

Therefore, the radius of the circle is 9 units.

 

Important Notes on Equation of Circle

These are a few things to remember when studying the equation of a circle

  • The general form of the equation of circle always has x² + y² in the beginning.
  • When a circle crosses two axes, then it has four points of intersection with those axes.
  • A circle that touches both axes has only two points of contact.
  • If any equation is of the form x²+y²+axy+C=0, then it is not the equation of the circle. There is no xy term in the equation of a circle.
  • The equation of circle always represents in polar form as r and θ.
  • The radius is the distance between the center and any point on the circle’s boundary. Therefore, the radius of a circle is always positive.

Frequently Asked Questions About Equation Of A Circle Calculator

What is the equation for a circle?

The equation for a circle is given by: (x-h)²+(y-k)² = a², Where (h,k) is the center and a is the radius of the circle.

What are the formulas for circles?

The circumference is equal to 2 (pi) of radius or pi of diameter. A circle’s area is equal to the square of its radius.

What is the equation of a circle when the center is at the origin?

At origin, the value of coordinates is (0,0), therefore, the equation of circle becomes:

(x-0)² + (y-0)² = r²

x² + y² = r²

If (x-4)²+(y+7)²=9 is the equation of a circle, then what is the center of the circle?

Given, (x-4)²+(y+7)²=9 is the equation of a circle. Comparing this equation with the standard equation, we get:

(x-h)²+(y-k)² = a²

h=4 and y = -7

Therefore, (4,-7) is the center of the circle.

How do we know if an equation is the equation of a circle?

If x and y are squared and the coefficient of x² and y² are the same, then it is an equation of the circle. For example, 3x²+3y² = 12 is a circle’s equation.