Equation of a Circle
(x − h)² + (y − k)² = r² has center (h, k) and radius r.
About this topic
A circle with centre (h, k) and radius r has the equation (x − h)² + (y − k)² = r², which is the distance formula in disguise. With centre at the origin it simplifies to x² + y² = r².
To find the centre and radius from an expanded equation, complete the square in both x and y. Points inside the circle make the left side smaller than r², and points outside make it larger, which is useful for describing ranges such as the coverage area of a radio tower.
Worked example: a circle has centre (2, −3) and radius 5, so its equation is (x − 2)² + (y + 3)² = 25. Given x² + y² − 6x + 4y − 12 = 0, group terms and complete the square: (x − 3)² + (y + 2)² = 12 + 9 + 4 = 25, so the centre is (3, −2) and r = 5. The point (6, 2) gives 9 + 16 = 25, so it lies on the circle centered at (3, −2).
Radio coverage, sonar ranges and orbit models use these equations. Two common slips are sign errors, since x + 3 means h is negative 3, and forgetting to take the square root of the right side to get the radius. When completing the square, add the same numbers to both sides. The equation x² + y² = 9 has radius 3, not 9. Compare a point's distance squared to r² to decide whether it lies inside, on or outside the circle.
Key ideas
- Derived from the distance formula
- Complete the square to find center
- Unit circle has r = 1
Quick check
Use cases: where Equation of a Circle helps
- Homework: review the key idea behind Equation of a Circle before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Equation of a Circle without notes.
- Visual learning: play with the interactive lab to see how Equation of a Circle changes when you adjust its inputs.
- Teaching a friend: explain Equation of a Circle in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Equation of a Circle?
(x − h)² + (y − k)² = r² has center (h, k) and radius r.
What will I learn in Equation of a Circle?
Derived from the distance formula. Complete the square to find center. Unit circle has r = 1.
Is Equation of a Circle part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Equation of a Circle lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Equation of a Circle?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.