Circles: Centre, Radius, and the Equation of a Circle

A circle is the set of every point that sits a fixed distance from one fixed point. That sentence is the whole definition, and remarkably, translating it directly into coordinates gives you the equation with no extra work.

The Standard Equation

• is the centre.
• is the radius.

Centred at the origin, and are both zero and the equation collapses to the familiar .

Where It Comes From

Take the centre and any point on the circle. The horizontal gap between them is ; the vertical gap is . Those two gaps form the legs of a right triangle whose hypotenuse is the radius.

Pythagoras gives immediately. It is the distance formula with the distance fixed at and both sides squared to clear the root.

Two consequences fall out of that derivation:

• The right side is , not . If the equation reads , the radius is .
• Squaring removes any sign problem. is the same whether the point is left or right of the centre, which is exactly what you want.

Reading Centre and Radius

The equation subtracts the centre coordinates, so reading one back means negating what you see.

A plus sign inside a bracket always means a negative coordinate. Rewriting as makes it obvious, and it is worth doing on paper until the habit sticks.

To sketch a circle quickly, plot the centre and then mark four points at distance directly up, down, left and right. Those four are enough to freehand a decent curve.

General Form and Completing the Square

Expanding the standard form scatters everything:

That second shape is general form, . You cannot read anything useful from it, so the job is to reverse the expansion by completing the square on each variable.

The procedure, in order:

1. Move the constant to the right side.
2. Group the -terms together and the -terms together.
3. Complete each square. For , add . Do the same for .
4. Add the same amounts to the right side. This is the step people skip, and skipping it changes the radius.
5. Factor each group into a squared bracket.

If the equation has a coefficient in front of and — say — divide the whole equation by it first. A genuine circle always has equal coefficients on and ; if they differ, the curve is an ellipse or a hyperbola, not a circle.

When It Is Not a Circle

After completing the square you have for some number . Three cases:

• : a genuine circle of radius .
• : a single point at . Sometimes called a degenerate circle.
• : no graph at all. Two real squares cannot add to a negative number.

Getting a negative on the right is usually a sign you forgot step 4 above — but not always, so check the arithmetic before assuming an error.

Finding the Equation From Given Information

Centre and radius given. Substitute straight in.

Centre and one point on the circle. The distance between them is the radius.

For centre through :

Endpoints of a diameter given. The centre is the midpoint, and the radius is half the diameter’s length.

Worked Example A: Read It Off

State the centre and radius of .

Centre ; radius .

Worked Example B: Complete the Square

Find the centre and radius of .

Half of 10 is 5, and . Half of is , and .

Centre , radius .

Worked Example C: From a Diameter

A circle has diameter endpoints and . Find its equation.

Centre is the midpoint: .

Diameter length: , so .

Worked Example D: Is a Point Inside or Outside?

Is inside, on, or outside ?

Substitute: .

Since , the point is inside the circle. Comparing the substituted value with is all this test ever requires: less means inside, equal means on, greater means outside.

Worked Example E: A Degenerate Case

What does look like?

The only way two squares sum to zero is if both are zero, so the “circle” is the single point .

Common Mistakes to Avoid

• Forgetting to square-root the right-hand side. means radius , not 36.
• Getting the centre’s signs backwards. means .
• Only adding to the left when completing the square. Whatever you add inside must be added on the right too.
• Halving before squaring, or squaring before halving. The rule is half first, then square: for , half is , squared is .
• Not dividing out a leading coefficient. must become first.
• Calling an unequal-coefficient equation a circle. is an ellipse.
• Assuming a negative right-hand side is an error. It can be, but it can also be a legitimate “no graph” answer.

Where Circles Lead Next

• Conic sections. Circles are the simplest conic; ellipses, parabolas and hyperbolas follow the same completing-the-square treatment.
• Tangent lines. A tangent meets the circle once and is perpendicular to the radius at the point of contact.
• Circumscribed circles. The centre of a circle through three points is where the perpendicular bisectors of the sides meet.
• Trigonometry. The unit circle is the foundation of sine and cosine.
• Distance-based modelling — signal range, safety radii, and collision detection all reduce to “is this point within ?”

Practice Problems

Work each one before opening the answer.

Problem 1. State the centre and radius of .

Problem 2. State the centre and radius of .

Problem 3. Write the equation of the circle with centre and radius .

Problem 4. Write the equation of the circle with centre and radius .

Problem 5. Find the centre and radius of .

Problem 6. Find the centre and radius of .

Problem 7. A circle has centre and passes through . Find its equation.

Problem 8. A diameter runs from to . Find the circle’s equation.

Problem 9. Is inside, on, or outside ?

Problem 10. What is the graph of ?

Problem 11. Convert to general form.

Problem 12. Find the centre and radius of .

Quick Reference

Circles extend the distance formula into a curve, and converting from general form relies on completing the square. Tangents to a circle use perpendicular slopes, and the plotting technique is the one from Graphing Equations. More Algebra lessons are available.