Equation of Circle: Formula, Examples

What is the Equation of Circle?

Equation of circle helps us show a circle on a graph using maths. It tells us where the circle is placed and how big it is. To draw a circle on the coordinate plane, we need to know two things: the coordinates of its center and the length of its radius. Once we know these, we can easily write the equation of circle. So, the answer to the commonly asked question 'What is the equation of circle?' is that it is simply a way to represent all the points that make up the circle. 

Every point on the circle remains at the same distance from its centre. This fixed distance is called the radius. So, the equation of circle shows the relation between the centre, radius, and all the points lying on the circumference of the circle. To learn more about the equation of circle and its different forms with examples, keep reading.

Read More: Segment of a Circle

Equation of Circle and Its Different Forms

Equation of circle helps us understand how a circle looks and where it is placed on a graph or coordinate plane. When we know the center of the circle and its radius, we can easily write its equation and draw it on the plane. In coordinate geometry, a circle can be written in different ways, and each form gives us useful information about the circle’s shape and position.

The main forms of the equation of circle are the general equation of a circle, the standard equation of a circle, the parametric equation of a circle, and the polar equation of a circle. Let’s understand each of these forms one by one.

General Equation of a Circle

General equation of a circle helps us find the center and radius of a circle when its equation is given in the form:

x² + y² + 2gx + 2fy + c = 0

• In this equation, g, f, and c are constants. From this, the center of the circle is (-g, -f) and the radius is √(g² + f² - c). 

• This form is useful in coordinate geometry because it tells us where the circle lies on the graph. 

• But, the general equation of a circle can be a bit difficult to understand, so we generally convert it into the standard equation of a circle using the completing the square method.

Standard Equation of a Circle

Standard equation of a circle helps us find the exact position of the circle on a graph. It clearly shows both the center and the radius of the circle. The standard equation of a circle with centerat (x₁, y₁) and radius r is: (x − x₁)² + (y − y₁)² = r²

• Here, (x, y) is any point on the circle, and the distance between this point and the center is always equal to the radius r.

• Using the distance formula: √((x − x₁)² + (y − y₁)²) = r

• Squaring both sides gives us the standard equation of a circle: (x − x₁)² + (y − y₁)² = r²

For example, if the equation of a circle is (x − 4)² + (y − 2)² = 36, then the center of the circle is (4, 2) and the radius is 6 units.

Polar Equation of a Circle

Polar equation of a circle shows how the position of any point on a circle can be represented using the distance from the origin (r) and the angle (θ). To understand it better, let’s take a circle whose center is at the origin and whose radius is p.

• The equation of a circle in Cartesian form is: x² + y² = p²

• Now, we know: x = rcosθ and y = rsinθ.

• Putting these in the circle equation:
  (rcosθ)² + (rsinθ)² = p²
  r²(cos²θ + sin²θ) = p²
  r²(1) = p²
  So, r = p

Hence, the polar equation of a circle centred at the origin is: r = p, where p is the radius of the circle. This means, for every point on the circle, the distance from the origin (r) remains the same as the radius (p).

Parametric Equation of a Circle

Parametric equation of a circle is another way to show the position of any point on a circle using an angle θ.

As we learnt, the general equation of a circle is: x² + y² + 2hx + 2ky + C = 0.

• Here, the center of the circle is (−h, −k) and the radius is r.

• If we take a point (x, y) on the boundary of the circle, and the line joining this point to the center makes an angle θ, then the parametric equation of a circle is given by:

• x = −h + rcosθ

• y = −k + rsinθ

These equations help us find any point on the circle using the radius and angle θ.

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Equation of a Circle Formula

The equation of a circle helps us find the relation between any point on the circle and its center. If the center of the circle is at (x₁, y₁) and the radius is r, then the equation of a circle formula is:

(x - x₁)² + (y - y₁)² = r²

Here,

• (x₁, y₁) = coordinates of the center

• r = radius of the circle

• (x, y) = any point on the circle

This equation shows that every point (x, y) on the circle is at a fixed distance r from the centre.

Read More: Perimeter of a Circle

How to Find Equation of Circle?

After learning what is equation of circle, another question that comes to students' minds is, 'How to find equation of circle?’ We can find the equation of a circle using different ways, depending on where the circle is positioned on the coordinate plane. We can use the center and radius of the circle to find its equation. Let’s understand step by step how to find the equation of circle in different cases.

Equation of Circle With Center at (x₁, y₁):

Take any point on the circle as (x, y). The distance between this point and the center (x₁, y₁) will always be equal to the radius (r). Using the distance formula, we will get: √((x - x₁)² + (y - y₁)²) = r

• Now, squaring both sides gives: (x - x₁)² + (y - y₁)² = r²

• This is the general form that works for any circle on the plane.

Equation of Circle With Center at the Origin (0, 0):

If the center of the circle is at the origin, then its coordinates are (0, 0). Take a point (x, y) on the circle. The distance between (x, y) and (0, 0) is equal to r.

• Using the distance formula: √((x - 0)² + (y - 0)²) = r

• After squaring both sides, we get: x² + y² = r²

Equation of Circle With Center on x-Axis

Suppose the circle’s center is on the x-axis at (a, 0). A point on the circle is (x, y), and the distance between them equals the radius (r).

• Using the distance formula: √((x - a)² + (y - 0)²) = r

• Squaring both sides, we get: (x - a)² + y² = r²

Equation of Circle With Center on Y-Axis

If the center is on the y-axis at (0, b), take a point (x, y) on the circle. The distance between them is r.

• Now use the distance formula: √((x - 0)² + (y - b)²) = r

• Then square both sides gives: x² + (y - b)² = r²

Equation of Circle Touching x-Axis

If a circle just touches the x-axis, it means the y-coordinate of its center equals the radius. So, the center is (a, r). Take a point (x, y) on the circle.

• Distance between them = r

• Using the formula: √((x - a)² + (y - r)²) = r

• Squaring both sides, we get: (x - a)² + (y - r)² = r²

Equation of Circle Touching y-Axis

If a circle touches the y-axis, the x-coordinate of its center is equal to the radius. So, the center is (r, b).

• Taking a point (x, y) on the circle and using the distance formula: √((x - r)² + (y - b)²) = r

• Squaring both sides gives: (x - r)² + (y - b)² = r²

Equation of Circle Touching Both X-axis and Y-axis:

If the circle touches both x- and the y-axis, then its center is at the same distance from both axes, that is, at (r, r). Take a point (x, y) on the circle.

• Using the distance formula,
  √((x - r)² + (y - r)²) = r

• By squaring both sides, to get:
  (x - r)² + (y - r)² = r²

Read More: Euler's Formula

Equation of Circle Examples With Solutions

After learning how to find the equation of circle using different formulas, it’s now time to see how to apply those formulas in actual questions. Solving examples helps you understand when and how to use the standard, general, or polar equation of a circle correctly. Let’s look at some solved equation of circle examples step by step.

Example 1: Find the equation of the circle in standard form if the center is (3, –2) and the radius is 4.

Solution:  We know that the standard equation of a circle is:
(x - x1)² + (y - y1)² = r²

Here, (x1, y1) = (3, –2) and r = 4.
Substitute the values in the formula:
(x - 3)² + [y - ( -2 )]² = 4²
(x - 3)² + (y + 2)² = 16

So, the required equation of the circle is:
(x - 3)² + (y + 2)² = 16

Example 2: Find the equation of a circle with center (–4, 1) and radius 5.

Solution: Using the standard equation of a circle:
(x - x1)² + (y - y1)² = r²

Substitute (x1, y1) = (–4, 1) and r = 5:
[x - ( -4 )]² + (y - 1)² = 5²
(x + 4)² + (y - 1)² = 25

Hence, the equation of the circle is:
(x + 4)² + (y - 1)² = 25

Example 3: Convert the equation x² + y² = 9 into its polar form.

Solution: We know that:
x = r cosθ and y = r sinθ

Substitute these values in the given equation:
(r cosθ)² + (r sinθ)² = 9
r² (cos²θ + sin²θ) = 9
r² (1) = 9
r = 3

So, the polar equation of the circle is:
r = 3

Also Read: Area of Semicircle

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