A circle is every point at a fixed distance from a centre. The distance formula turns that sentence into an equation.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Standard form
(x − h)² + (y − k)² = r², with centre (h, k) and radius r.
The signs reverse
In (x − 3)² the centre coordinate is 3, and in (x + 3)² it is −3. The form subtracts, so the sign appears flipped.
The right side is r squared
An equation ending in 16 means a radius of 4, not 16. Take the square root before reporting it.
General form
An expanded equation hides the centre. Completing the square on both variables recovers standard form.
When it is not a circle
A right side of zero gives a single point, and a negative right side gives no graph at all.
The equation is the distance formula
(x − h)² + (y − k)² = r² states that every point on the circle is distance r from the centre. Squaring both sides of the distance formula produces it directly.
Reading centre and radius
The centre is (h, k) and the radius is the square root of the right-hand side. The minus signs inside mean the coordinates appear with their signs flipped, which is easy to misread.
Completing the square reveals it
A sprawling equation like x² + y² − 6x + 4y = 12 hides its centre. Completing the square in x and in y separately converts it into recognisable form.
Testing whether a point lies on the circle
Substitute the coordinates. Equal to r² means on the circle, less means inside, greater means outside. The equation classifies every point in the plane, not only those on the curve.
Step 2: Try It Yourself
Tap and try it out.
- Diameter10 cm
- Circumference31.42 cm
- Circumference ÷ diameter3.14
Change the size. The circle gets bigger, but circumference divided by diameter stays at about 3.14 every time. That number is π.
Step 3: Watch an Example
One step at a time.
Watch Kofi Recover a Centre
Kofi is given x² + y² − 6x + 4y − 3 = 0.
- Step 1
He groups the terms: (x² − 6x) + (y² + 4y) = 3.
Step 4: Your Turn
Practice makes it stick.
The Radius
Problem 1 of 2
(x − 2)² + (y − 5)² = 49. What is the radius?
The Centre
Problem 2 of 2
Same circle. What is the x-coordinate of the centre?
Read the Circle
1 of 8
x² + y² = 36. What is the radius?
2 of 8
(x − 4)² + y² = 25. Radius?
3 of 8
Same circle. Centre x-coordinate?
4 of 8
(x + 5)² + (y − 1)² = 9. Centre x-coordinate?
5 of 8
To complete the square on x² − 6x, what is added?
6 of 8
x² + y² = 0. How many points does the graph contain?
7 of 8
Put the steps for finding a centre from general form in order.
- 1Move the constant to the right.
- 2Complete the square on each group, adding to both sides.
- 3Read the centre and take the root of the right side.
- 4Group the x terms and the y terms.
8 of 8
(x − 1)² + (y + 3)² = 64. Radius?
Step 5: Quick Check
Show what you know.
Question 1 of 2
(x − 3)² + (y − 2)² = 16. What is the radius?
Question 2 of 2
Where does the circle equation come from?
What You Learned
- A circle is (x − h)² + (y − k)² = r², with centre (h, k).
- The right side is r squared, so take the root for the radius.
- Completing the square recovers the centre from an expanded equation.