# Standard Equation of a Circle Calculator

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:

where:

- $(x,y)$ - The
**coordinates**of any**point**on the circle; - $(a,b)$ - The
**coordinates**of the**center**of the circle; and - $r$ - 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:

where:

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

It is a straightforward conversion between these two forms; the additional parameter $\alpha$ 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:

where:

- $(x,y)$ - The
**coordinates**of any**point**on the circle; - $D$ - The
**sum**of the**coefficients**of the**x-terms**; - $E$ - The
**sum**of the**coefficients**of the**y-terms**; and - $F$ - 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:

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**.

## Other related calculators to find the equation of a circle

- Equation of a circle calculator;
- Center of a circle calculator;
- General form of the equation of a circle calculator;
- Equation of a circle with diameter endpoints calculator;
- General to standard form of a circle calculator;
- Standard form to general form of a circle calculator.

## 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:

**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`

.**Substitute**the value of**radius**in the place of`r`

in this equation. This gives`x²+y² = 7²`

.**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:

**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.

- If