# What is the equation of a circle? A Comprehensive Guide

Defining a circle in simple terms is pretty simple because it is a regular shape having a fixed point from where all points on its boundary have the same distance. But explaining this specific figure of geometry is pretty difficult, especially in short form. This is where the** equation of a circle **becomes important and is considered by Mathematics students.

It is a general equation that helps in explaining a circle in a few lines and quickly. The main advantage of this equation is that it enables a student to check whether a point lies on the circumference of a circle or not. In short, using an equation of a circle can be the easiest approach to defining a circle and its related po

## What is the standard equation of a circle?

As mentioned, the equation of a circle explains a circle in terms of its parameters like radius, circumference, and coordinates of origin. That’s why, this equation can be written in different formats one of which is the **standard equation of a circle**.

It is also considered the simples equation of a circle because of the involvement of a few coordinates. Here is the standard equation of a circle:

**(x - h) ^{2} + (y - k)^{2} = r^{2}**

Here, **“h”** and **“k” **are the coordinates of the origin/center of the circle while **“r”** represents the radius of the circle. Using this equation, one can write the equation of a circle having any measure of radius.

Undoubtedly, it is the simplest form of the equation of a circle and we can convert it into other forms like a general form of the equation of a circle. To convert the standard equation of a circle to its general form, we only have to put the values of** “h”, “k”,** and **“r”**.

Yes, it is pretty simple to convert the standard equation to its general form if you know these parameters. You only have to put those parameters in the above equation and solve it to find the general form of the circle.

## What is the radius of a circle equation?

To understand the** radius** and other parameters of the circle, you need to learn the definition of this figure first. A circle is a specific two-dimensional figure having a fixed point that is called its center and every point on its boundary is equidistant from that fixed point.

The radius is the distance of any point on its boundary from its center. In terms of this parameter, a specific equation can be written that is called the radius of a circle equation. This equation depends on the radius of the circle.

Using this specific equation, you can find the equation of a circle just by inserting the value of the radius. Here is the general form of this equation.

**x ^{2 }+ y^{2} = r^{2}**

As you can see only missing measurement in this equation is the radius. We can write a general equation of a circle with a specific radius measurement using this equation and a center at the origin where the coordinates will be **(0, 0)**. So, the values for **“h”** and **“k”** will be** “0”.**

## What is the circumference of a circle equation?

Circumference of a circle is its boundary that is made by the continuous points joining with the same distance from its center. We can find this parameter using the radius of the circle. Here is the formula you can use for this calculation.

**Circumference** **= 2𝝿r**

As this parameter depends on the radius, we can write the equation for the circumference of a circle. Here is the modified form of the equation of a circle depending on the circumference instead of the radius.

**x ^{2} + y^{2}**

**= (C2𝝿)**

^{2}In this equation, “c” represents the circumference of the circle while “𝝿” is the constant number showing a specific ratio between the circumference and radius of the circle. Its value remains the same for all circles and in all conditions.

## What is the equation for the area of a circle?

As mentioned earlier, a circle is a 2D figure with a plane interface. So, we can find its area too using a specific formula. The equation for the area of a circle depends on the measurement of radius only as it remains the same from the center of the circle.

Here is the equation/formula using which we can find the area of a circle.

**Area of a Circle** = **𝝿r ^{2}**

Doesn’t matter whether you have a small circle or a large one, you can easily find its area using the above formula.

## How to find the equation of a circle?

Finding the equation of a circle is pretty simple as you only need to know the values of the **coordinates** of the center and the measurement of the radius. You can use the following equation of a circle formula to get the equation using the above parameters.

**(x - h) ^{2 }+ (y - k)^{2} = r^{2}**

If you know basic mathematical operations, you can do this calculation easily. Sometimes, you may have a short time to find this equation for the completion of your assignment. You can get assistance from our** Equation of a Circle Calculator.**

This **online tool **will help you in finding the equation of a circle in different formats including standard, general, and parametric. Yes, you can get the equation for this figure in different formats to solve the related problems.

## A practical example of calculating the equation of a circle

To assist our readers in becoming proficient in finding the equation of a circle, we have solved an example here.

**example 1**:

Find the standard equation of a circle having a center at (**3, 7**) and the radius is **8m**.

**Solution**:

As we know the standard equation of a circle, we have to use these parameters in that equation.

**(x - h) ^{2} + (y - k)^{2} = r^{2}**

So, we have to put the given values in this** formula** only.

**(x - 3) ^{2} + (y - 7)^{2} = (8)^{2}**

**x ^{2} + 9 - 6x + y^{2} + 49 - 14y = 64**

By solving this equation, we get:

**x ^{2} - 6x + y^{2} - 14y + 58 - 64 = 0**

So, the standard equation of the circle having the above parameters will be written as:

**x ^{2} - 6x + y^{2} - 14y - 6 = 0**

## Conclusion

In the above blog, we have discussed the equation of a circle in detail with the derivation of this equation in different forms. You may have learned from this blog and are ready to solve problems related to this equation/figure manually.

If you are dealing with a short deadline and want to learn more about this equation, you can get assistance from our **Equation of a Circle Calculator**. It will enable you to learn a lot without investing much time.

**FAQ**

**FAQ**

**What is the radius of a circle whose equation is ****x2 + y2 + 8x -6y +21 = 0****?**

The radius of a circle having the above equation will be **“5”** units.

**What is the radius of a circle whose equation is** **x ^{2 }+ y^{2} - 10x +6y +18 = 0**

**?**

After simplifying this equation, the radius of the circle comes out to be **“4”** units.

**What is the center of a circle represented by the equation** **(x + 9) ^{2 }+ (y - 6)^{2} = (10)^{2}**

**?**

The equation of a circle is written in the following format:

**(x - h) ^{2} + (y - k)^{2} = r^{2}**

In this equation,** “h”** and “k” represents the coordinates of the center of the circle. By comparing this standard equation with the given equation, the coordinates for the center of the circle will be **(-9, 6)**.

**What is the standard form of the equation of the circle with a center**** (-2, -3) and a radius of 24****?**

The standard equation of a circle having the above parameters will be,

**(x + 2) ^{2} + (y + 3)^{2} = (24)^{2}**

**The diameter of a circle is 4 cm. Which equation can be used to find its circumference?**

To find the circumference of the circle while the diameter is given, we can use the following formula.

**Circumference** = 𝝿d

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

The equation of a circle will be simplified to the following form when the center is at the origin:

**x ^{2} + y^{2} = r^{2}**

**What is C in the General Equation of a Circle?**

Sometimes, the** coordinates** of the center and the value of the radius are missing in the question. **“C” **is the constant term used to determine the values of these parameters.

**What is the General Equation of a Circle?**

The general equation of a circle is written as

**x ^{2 }+ y^{2} = r^{2}**