Circle Equations: Standard, General, Diameter, and Parametric Forms

Circle equations feel confusing when every form looks like a separate formula.

But a circle is not four different things. It is one simple idea written in different languages.

A circle is the set of all points that stay at the same distance from one fixed point. The fixed point is the centre. The fixed distance is the radius.

Once you hold on to that idea, the forms become much easier:

• standard form tells you the centre and radius directly
• general form hides the centre and radius inside expanded algebra
• diameter form starts from the two ends of a diameter
• parametric form traces the circle point by point

This guide will help you choose the right form quickly, convert between forms, and avoid the small sign mistakes that usually cost marks.

The One Idea Behind Every Circle Equation

Imagine fixing one point on a coordinate plane. Now place a compass needle on that point and rotate the pencil without changing its opening. Every point touched by the pencil is at the same distance from the centre.

That distance is the radius.

If the centre is (h, k) and any point on the circle is (x, y), then the distance from (h, k) to (x, y) is always r.

Using the distance formula:

sqrt((x - h)^2 + (y - k)^2) = r

Squaring both sides gives:

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

This is the heart of the whole topic.

Quick Decision Guide: Which Form Should You Use?

Before solving, read the given information and choose the form from the clues.

What the question gives	Best form to start with	Why it helps
Centre and radius	Standard form	Direct substitution
Centre and one point on the circle	Standard form	Find radius by distance formula
Expanded equation	General form	Complete the square
Endpoints of a diameter	Diameter form or midpoint method	Centre and radius come from the endpoints
A point moving around a circle	Parametric form	Uses cos t and sin t
Need to sketch the circle	Standard form	Centre and radius are visible
Need final answer as an expanded equation	General form	Expand at the end

Standard Form: The Most Useful Form for Understanding the Circle

The standard form of a circle is:

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

Here:

• centre is (h, k)
• radius is r
• r^2 is the number on the right side

If the centre is the origin (0, 0), the equation becomes:

x^2 + y^2 = r^2

Example 1: Centre and Radius Are Given

Find the equation of the circle with centre (3, -2) and radius 5.

Use:

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

Here h = 3, k = -2, and r = 5.

So:

(x - 3)^2 + (y - (-2))^2 = 5^2

Therefore:

(x - 3)^2 + (y + 2)^2 = 25

Notice the sign carefully. If the centre has -2 as the y-coordinate, the bracket becomes y + 2.

Example 2: Centre and a Point Are Given

Find the equation of the circle with centre (1, 4) passing through (4, 8).

The radius is the distance from the centre to the given point.

r = sqrt((4 - 1)^2 + (8 - 4)^2)

r = sqrt(3^2 + 4^2)

r = 5

Now use standard form:

(x - 1)^2 + (y - 4)^2 = 25

This is why centre-and-point questions are usually standard-form questions in disguise.

General Form: The Expanded Form That Hides the Circle

The general form is commonly written as:

x^2 + y^2 + 2gx + 2fy + c = 0

If it represents a real circle, then:

centre = (-g, -f)
radius = sqrt(g^2 + f^2 - c)

But do not rush to use this formula blindly.

First check three things:

1. The coefficients of x^2 and y^2 should be equal.
2. There should be no xy term.
3. If the equal coefficient is not 1, divide the whole equation by that coefficient first.

For example:

2x^2 + 2y^2 - 8x + 12y - 6 = 0

This can represent a circle, but first divide by 2:

x^2 + y^2 - 4x + 6y - 3 = 0

Only then should you identify 2g, 2f, and c.

Completing the Square: The Safer Way to Handle General Form

Even if you know the direct formula, completing the square is often safer because it shows the centre and radius clearly.

Example 3: Convert General Form to Standard Form

Find the centre and radius of:

x^2 + y^2 - 6x + 4y - 12 = 0

Step 1: Group the x terms and y terms.

(x^2 - 6x) + (y^2 + 4y) = 12

Step 2: Complete the square for each group.

For x^2 - 6x, half of -6 is -3, and (-3)^2 = 9.

For y^2 + 4y, half of 4 is 2, and 2^2 = 4.

Add the same numbers to both sides:

(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4

Step 3: Write each group as a square.

(x - 3)^2 + (y + 2)^2 = 25

Now read the answer:

centre = (3, -2)
radius = 5

The original equation looked busy, but the circle was simply hiding inside it.

Standard Form vs General Form

Students often ask which form is “better”. The answer depends on the job.

Purpose	Prefer this form
Find centre and radius	Standard form
Sketch the circle	Standard form
Substitute centre and radius quickly	Standard form
Match an expanded answer option	General form
Convert from a mixed equation	Complete square into standard form
Compare algebraic equations	General form

Here is the same circle in both forms:

(x - 3)^2 + (y + 2)^2 = 25

Expand:

x^2 - 6x + 9 + y^2 + 4y + 4 = 25

Bring everything to the left:

x^2 + y^2 - 6x + 4y - 12 = 0

Both equations describe the same circle.

The standard form is easier to understand. The general form is easier to match with many answer choices.

Diameter Form: When Two Opposite Ends Are Given

If a question gives the endpoints of a diameter, you can solve it in two ways.

Let the endpoints be:

A(x1, y1) and B(x2, y2)

Method 1: Midpoint and Radius

The centre is the midpoint of the diameter:

centre = ((x1 + x2)/2, (y1 + y2)/2)

The radius is half the length of the diameter:

radius = AB/2

This method is the most natural because it returns you to standard form.

Method 2: Direct Diameter Form

The direct diameter form is:

(x - x1)(x - x2) + (y - y1)(y - y2) = 0

This works because the angle made at any point on a circle by the two ends of a diameter is a right angle.

If this idea feels abstract, use Method 1. It is usually easier to remember and easier to check.

Example 4: Endpoints of Diameter Are Given

Find the equation of the circle whose diameter has endpoints (1, 2) and (5, -4).

First find the centre:

centre = ((1 + 5)/2, (2 + (-4))/2)

centre = (3, -1)

Now find the length of the diameter:

AB = sqrt((5 - 1)^2 + (-4 - 2)^2)

AB = sqrt(4^2 + (-6)^2)

AB = sqrt(52)

So:

radius = sqrt(52)/2 = sqrt(13)

Now write the standard form:

(x - 3)^2 + (y + 1)^2 = 13

If expanded, this becomes:

x^2 + y^2 - 6x + 2y - 3 = 0

Now check with direct diameter form:

(x - 1)(x - 5) + (y - 2)(y + 4) = 0

Expand:

x^2 - 6x + 5 + y^2 + 2y - 8 = 0

x^2 + y^2 - 6x + 2y - 3 = 0

Both methods give the same answer.

Parametric Form: A Circle as a Moving Point

Standard and general forms describe the full circle at once.

Parametric form describes a point moving around the circle.

For a circle with centre (h, k) and radius r, the parametric form is:

x = h + r cos t
y = k + r sin t

For one full round:

0 <= t <= 2pi

Here, t is the parameter. You can think of it as the angle that controls where the moving point is on the circle.

Why Cos and Sin Appear

For a circle centred at the origin with radius r, a point on the circle can be written as:

(r cos t, r sin t)

This works because:

cos^2 t + sin^2 t = 1

So:

x^2 + y^2 = (r cos t)^2 + (r sin t)^2

x^2 + y^2 = r^2(cos^2 t + sin^2 t)

x^2 + y^2 = r^2

If the centre shifts from (0, 0) to (h, k), add h to the x-coordinate and k to the y-coordinate.

That gives:

x = h + r cos t
y = k + r sin t

Example 5: Write Parametric Form

Write the parametric equations of:

(x - 3)^2 + (y + 2)^2 = 25

The centre is (3, -2) and the radius is 5.

So:

x = 3 + 5 cos t
y = -2 + 5 sin t

where:

0 <= t <= 2pi

Converting Parametric Form Back to Standard Form

Suppose:

x = 3 + 5 cos t
y = -2 + 5 sin t

Move the centre parts:

x - 3 = 5 cos t
y + 2 = 5 sin t

Divide by 5:

(x - 3)/5 = cos t
(y + 2)/5 = sin t

Now square and add:

((x - 3)/5)^2 + ((y + 2)/5)^2 = cos^2 t + sin^2 t

Since cos^2 t + sin^2 t = 1:

(x - 3)^2/25 + (y + 2)^2/25 = 1

Multiply by 25:

(x - 3)^2 + (y + 2)^2 = 25

So the parametric form and standard form are just two ways to describe the same circle.

A Practical Comparison of All Four Forms

Use this table for revision before solving practice questions.

Form	Equation pattern	Best use
Origin-centre form	x^2 + y^2 = r^2	Centre is (0, 0)
Standard form	(x - h)^2 + (y - k)^2 = r^2	Centre and radius are needed
General form	x^2 + y^2 + 2gx + 2fy + c = 0	Expanded equation is given
Diameter form	(x - x1)(x - x2) + (y - y1)(y - y2) = 0	Endpoints of diameter are given
Parametric form	x = h + r cos t, y = k + r sin t	A point moves around the circle

Common Mistakes in Circle Equations

Circle equations are not difficult, but they are very easy to spoil through small sign errors.

Mistake 1: Reading the Centre With the Same Signs

If the equation is:

(x - 4)^2 + (y + 7)^2 = 36

The centre is:

(4, -7)

It is not (-4, 7).

In standard form, the bracket signs are opposite to the coordinates.

Mistake 2: Taking the Right Side as Radius Instead of Radius Squared

If:

(x + 1)^2 + (y - 5)^2 = 49

The radius is:

7

not 49.

The right side is r^2, not r.

Mistake 3: Forgetting to Divide When x^2 and y^2 Have Equal Coefficients

If:

3x^2 + 3y^2 - 12x + 18y - 9 = 0

Do not treat -12 and 18 directly as 2g and 2f.

First divide by 3:

x^2 + y^2 - 4x + 6y - 3 = 0

Now work from the simpler equation.

Mistake 4: Calling Every Second-Degree Equation a Circle

This is not a circle in the usual coordinate axes:

x^2 + 4y^2 = 16

The coefficients of x^2 and y^2 are not equal.

This is also not a usual circle equation:

x^2 + y^2 + 3xy - 2x + 5y = 0

It has an xy term.

For the school-level circle form, look for equal squared coefficients and no xy term.

Mistake 5: Using Diameter as Radius

If the endpoints of a diameter are given, the distance between them is the diameter, not the radius.

Always halve it.

Better still, find the midpoint first, then calculate the radius from the midpoint to either endpoint.

A Clean Solving Routine

Use this routine whenever a circle question looks messy.

1. Check what is given.
2. Choose the starting form.
3. If the equation is expanded, collect x terms and y terms separately.
4. Complete the square if you need centre and radius.
5. Read the centre carefully by reversing bracket signs.
6. Take the square root of the right side to get the radius.
7. Expand only if the final answer asks for general form.

Mixed Practice With Solutions

Question 1

Find the centre and radius:

(x + 6)^2 + (y - 1)^2 = 16

Answer:

centre = (-6, 1)
radius = 4

The signs inside the brackets reverse for the centre, and sqrt(16) = 4.

Question 2

Convert to standard form:

x^2 + y^2 + 10x - 8y + 32 = 0

Solution:

(x^2 + 10x) + (y^2 - 8y) = -32

Complete the squares:

(x^2 + 10x + 25) + (y^2 - 8y + 16) = -32 + 25 + 16

So:

(x + 5)^2 + (y - 4)^2 = 9

Therefore:

centre = (-5, 4)
radius = 3

Question 3

Write the equation of the circle with centre (-2, 3) passing through (2, 6).

Find the radius:

r = sqrt((2 - (-2))^2 + (6 - 3)^2)

r = sqrt(4^2 + 3^2) = 5

Equation:

(x + 2)^2 + (y - 3)^2 = 25

Question 4

Write the equation of the circle whose diameter endpoints are (-1, 4) and (5, 2).

Centre:

((-1 + 5)/2, (4 + 2)/2) = (2, 3)

Diameter length:

sqrt((5 - (-1))^2 + (2 - 4)^2)

sqrt(6^2 + (-2)^2) = sqrt(40)

Radius:

sqrt(40)/2 = sqrt(10)

Equation:

(x - 2)^2 + (y - 3)^2 = 10

Question 5

Write the parametric form of:

(x + 4)^2 + (y - 1)^2 = 36

Centre is (-4, 1) and radius is 6.

So:

x = -4 + 6 cos t
y = 1 + 6 sin t

for:

0 <= t <= 2pi

Frequently Asked Questions

1. What is the easiest form of a circle equation?

Standard form is usually the easiest:

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

It shows the centre and radius directly, which makes sketching and checking much easier.

2. How do I find the centre from general form?

Convert the equation to standard form by completing the square. If the equation is already in the form x^2 + y^2 + 2gx + 2fy + c = 0, then the centre is (-g, -f).

3. Why do the signs reverse in standard form?

Because standard form uses (x - h) and (y - k). So if you see (x + 3), it means (x - (-3)), and the x-coordinate of the centre is -3.

4. Is the number on the right side the radius?

Not directly. In standard form, the right side is r^2. If the right side is 64, the radius is 8.

5. When should I use the diameter formula?

Use it when the two endpoints of a diameter are given. If you remember the direct formula, you can use it. If not, find the midpoint for the centre and half the distance for the radius.

6. What is the parametric form of a circle?

For centre (h, k) and radius r, the parametric form is:

x = h + r cos t
y = k + r sin t

It describes a point moving around the circle.

7. How do I know if a second-degree equation represents a circle?

For the usual circle form, the coefficients of x^2 and y^2 should be equal, and there should be no xy term. If the equal coefficient is not 1, divide the whole equation first.

8. Should I leave my answer in standard form or general form?

Leave it in the form the question asks for. If nothing is specified, standard form is usually clearer because it shows the centre and radius. If answer options are expanded, convert to general form.