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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.

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.

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.

**Standard Equation of a Circle Calculator:**

The standard equation of a circle is given by the formula (x – h)^2 + (y – k)^2 = r^2, where (h, k) represents the coordinates of the center of the circle and r represents the radius. The Standard Equation of a Circle Calculator allows you to calculate the equation of a circle when you know the center and radius values.

Example:

Let’s say you want to find the standard equation of a circle with a center at (3, -2) and a radius of 5. By using the Standard Equation of a Circle Calculator, you would input the center coordinates (3, -2) and the radius value of 5. The calculator would then perform the necessary calculations and provide you with the standard equation: (x – 3)^2 + (y + 2)^2 = 25.

**Equation of a Circle Calculator Given Two Points:**

If you have the coordinates of two points on a circle, you can find the equation of the circle. The Equation of a Circle Calculator Given Two Points helps you determine the equation of a circle using these two points.

Example:

Suppose you have two points A(1, 2) and B(4, 5) that lie on a circle. By using the Equation of a Circle Calculator Given Two Points, you would input the coordinates of the two points (1, 2) and (4, 5). The calculator will then calculate the center and radius of the circle and provide you with the equation of the circle.

**Standard Form Equation of a Circle Calculator:**

The standard form equation of a circle is written as x^2 + y^2 + Dx + Ey + F = 0, where D, E, and F are constants. The Standard Form Equation of a Circle Calculator allows you to convert the equation of a circle from the general form to the standard form.

Example:

Let’s say you have the equation of a circle given as 2x^2 + 2y^2 – 4x + 8y + 5 = 0. By using the Standard Form Equation of a Circle Calculator, you would input the coefficients of the equation (2, 2, -4, 8, 5). The calculator will then convert the equation to standard form, which in this case would be x^2 + y^2 – 2x + 4y + 5/2 = 0.

**Equation of a Circle in Standard Form Calculator:**

The Equation of a Circle in Standard Form Calculator is used to find the equation of a circle when it is given in standard form. It helps you determine the center, radius, and other properties of the circle.

Example:

Suppose you have the equation of a circle given in standard form as x^2 + y^2 – 6x + 8y – 9 = 0. By using the Equation of a Circle in Standard Form Calculator, you would input the coefficients of the equation (1, 1, -6, 8, -9). The calculator will then calculate the center, radius, and provide you with the equation of the circle.

**Standard Form of the Equation of a Circle Calculator:**

The Standard Form of the Equation of a Circle Calculator is used to convert the equation of a circle from the standard form to the general form. The general form of the equation of a circle is x^2 + y^2 + Dx + Ey + F = 0.

Example:

Let’s say you have the equation of a circle in standard form given as x^2 + y^2 + 4x – 6y – 9 = 0. By using the Standard Form of the

Equation of a Circle Calculator, you would input the coefficients of the equation (1, 1, 4, -6, -9). The calculator will then convert the equation to general form, which in this case would be x^2 + y^2 + 4x – 6y – 9 = 0.

**Equation of a Circle Calculator Given Center and Radius:**

If you know the coordinates of the center of a circle and its radius, the Equation of a Circle Calculator Given Center and Radius can help you find the equation of the circle.

Example:

Suppose you want to find the equation of a circle with a center at (-2, 3) and a radius of 6. By using the Equation of a Circle Calculator Given Center and Radius, you would input the center coordinates (-2, 3) and the radius value of 6. The calculator would then calculate the equation of the circle, which would be (x + 2)^2 + (y – 3)^2 = 36.

**Equation of a Circle Calculator Given Three Points:**

The Equation of a Circle Calculator Given Three Points allows you to determine the equation of a circle when you have the coordinates of three non-collinear points on the circle.

Example:

Suppose you have three points A(1, 2), B(4, 5), and C(6, -1) that lie on a circle. By using the Equation of a Circle Calculator Given Three Points, you would input the coordinates of the three points (1, 2), (4, 5), and (6, -1). The calculator will then calculate the center and radius of the circle and provide you with the equation of the circle.

**Equation of a Circle Calculator Diameter:**

The Equation of a Circle Calculator Diameter helps you find the equation of a circle when you know the diameter of the circle.

Example:

Let’s say you want to find the equation of a circle with a diameter of 10 units. By using the Equation of a Circle Calculator Diameter, you would input the diameter value of 10. The calculator would then calculate the equation of the circle, which would be x^2 + y^2 = 25.

**Standard Equation of a Circle:**

The standard equation of a circle, as mentioned earlier, is given by the formula (x – h)^2 + (y – k)^2 = r^2. It represents a circle with center (h, k) and radius r.

Example:

The standard equation of a circle with a center at (2, -3) and a radius of 4 units would be (x – 2)^2 + (y + 3)^2 = 16.

**Standard Form to General Form of a Circle Calculator:**

The Standard Form to General Form of a Circle Calculator is used to convert the equation of a circle from the standard form to the general form.

Example:

Suppose you have the equation of a circle in standard form as x^2 + y^2 – 4x + 6y + 9 = 0. By using the Standard Form to General Form of a Circle Calculator, you would input the coefficients of the equation (1, 1, -4, 6, 9). The calculator will then convert the equation to general form, which would be x^2 + y^2 – 4x + 6y + 9 = 0.

**Find Center and Radius of Circle:**

The Find Center and Radius of Circle calculator helps you determine the center and radius of a circle when given the equation of the circle.

Example:

Suppose you have the equation of a circle as (x + 3)^2 + (y – 2)^2 = 25. By using the Find Center and Radius of Circle calculator, it will analyze the equation and provide you with the center (-3, 2) and radius 5.

**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.

**Find the Equation of a Circle Whose Diameter Has Endpoints Calculator?**

To find the equation of a circle when you know the endpoints of its diameter, you can follow these steps:

Step 1: Find the midpoint of the diameter by averaging the x-coordinates and y-coordinates of the endpoints. Let’s say the endpoints of the diameter are (x1, y1) and (x2, y2), then the midpoint can be calculated as ( (x1 + x2)/2 , (y1 + y2)/2 ).

Step 2: Calculate the distance between one endpoint and the midpoint. This will give you the radius of the circle. The distance formula is √((x2 – x1)^2 + (y2 – y1)^2).

Step 3: Use the midpoint coordinates and the radius to write the equation of the circle in standard form ( (x – h)^2 + (y – k)^2 = r^2 ), where (h, k) represents the coordinates of the midpoint and r represents the radius.

Example:

Suppose the endpoints of the diameter are A(1, 2) and B(5, 6). To find the equation of the circle, we follow the steps above:

Step 1: The midpoint coordinates are ((1 + 5)/2, (2 + 6)/2) = (3, 4).

Step 2: The distance between one endpoint and the midpoint is √((5 – 3)^2 + (6 – 4)^2) = √8 = 2√2. This is the radius of the circle.

Step 3: The equation of the circle in standard form is (x – 3)^2 + (y – 4)^2 = (2√2)^2 = 8.

**How to Find the Radius of a Circle from an Equation Calculator?**

To find the radius of a circle from its equation, you need to identify the values of the coefficients in the equation. The general equation of a circle is (x – h)^2 + (y – k)^2 = r^2, where (h, k) represents the coordinates of the center and r represents the radius.

Example:

Suppose you have the equation of a circle given as (x – 2)^2 + (y + 3)^2 = 25. From this equation, you can identify the center coordinates as (2, -3) and the radius as √25 = 5.

**How to Find the Equation of a Circle Calculator?**

To find the equation of a circle, you need to have information about the center and radius of the circle. You can use various methods, such as knowing the center and radius, having the diameter endpoints, or having three non-collinear points on the circle.

Depending on the information available, you can use the appropriate equation of a circle formula and input the required values to find the equation of the circle.

**What is the Equation of a Circle with Center Calculator?**

The equation of a circle with center calculator helps you find the equation of a circle when you know the coordinates of its center. By inputting the center coordinates into the calculator, it performs the necessary calculations and provides you with the equation of the circle.

**How Do I Find the Equation of a Circle?**

To find the equation of a circle, you need to have information about its center and radius. If you have the center coordinates (h, k) and the radius value r, you can use the standard form equation of a circle, which is (x – h)^2 + (y – k)^2 = r^2.

If you have other information such as the diameter endpoints or

three non-collinear points on the circle, you can use specific formulas or methods to derive the equation of the circle.

**What is the Circle Equation Answers?**

The circle equation refers to the mathematical representation of a circle. The general form of the circle equation is (x – h)^2 + (y – k)^2 = r^2, where (h, k) represents the coordinates of the center of the circle and r represents the radius.

**What is the Equation of the Circle Online?**

The equation of the circle online refers to various online tools and calculators available that can help you find the equation of a circle based on the given information, such as the center, radius, diameter endpoints, or three points on the circle. These online tools can perform the necessary calculations and provide you with the equation of the circle.

**What is the Formula of a Circle Problems?**

The formula of a circle involves the use of specific equations and formulas to calculate various properties of a circle, such as the circumference, area, radius, or diameter.

Some common formulas related to circles include:

– Circumference of a Circle: C = 2πr, where C represents the circumference and r represents the radius.

– Area of a Circle: A = πr^2, where A represents the area and r represents the radius.

– Diameter of a Circle: d = 2r, where d represents the diameter and r represents the radius.

These formulas are used to solve problems related to circles in geometry and mathematics.