--- title: "The Equation of a Circle" description: "Read a circle's centre and radius from (x - a)² + (y - b)² = r², and recover them from an expanded equation by completing the square in both variables." canonical: https://lightmysky.com/learn/mathematics/the-equation-of-a-circle-mt_YARqmi_ioe source: https://lightmysky.com/learn/mathematics/the-equation-of-a-circle-mt_YARqmi_ioe.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # The Equation of a Circle Read a circle's centre and radius from (x - a)² + (y - b)² = r², and recover them from an expanded equation by completing the square in both variables. Subject: Mathematics · Area: Geometry · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/the-equation-of-a-circle-mt_YARqmi_ioe ## Ready when they can - Write the equation of the circle with centre (3, -1) and radius 5 - Find the centre and radius of x² + y² - 6x + 4y - 12 = 0 - Decide whether a given point is inside, on, or outside a circle ## Lesson: The Equation of a Circle Last stop you measured the gap between two points with Pythagoras. A circle is what you get when you fix that gap and let the second point roam. The Ridgeway bowl is a rim 6 m from a peg on the plan, so every point of the rim is 6 m from that peg and no point off the rim is. Written out with last stop's formula, the square root of (x - a)² + (y - b)² equals r. Squaring both sides clears the root and leaves (x - a)² + (y - b)² = r², which is the equation of the circle. Read that form carefully, because it is built to be read. The centre is (a, b), and each bracket carries the opposite sign of its coordinate, so a centre at (3, -1) gives (x - 3)² + (y + 1)². The right-hand side is r², not r, so a 25 there means a radius of 5. The diameter is twice the radius, so 10. A centre at the origin drops both brackets and leaves x² + y² = r². Nothing else may sit in front: in this form both squared terms have coefficient 1. *(drawing: Two brackets for the centre, one number for the radius squared.)* **Example.** Write the circle with centre (3, -1) and radius 5. Flip each coordinate's sign inside its own bracket: (x - 3)² + (y + 1)². The radius goes on the right, squared, so the equation is (x - 3)² + (y + 1)² = 25. Expanding shows the other face of the same circle: x² - 6x + 9 + y² + 2y + 1 = 25, which tidies to x² + y² - 6x + 2y - 15 = 0. That second form hides the centre and the radius completely, which is why the next step matters. **Example.** Now go the other way. Take x² + y² - 6x + 4y - 12 = 0 and complete the square in each variable separately. For x, x² - 6x is (x - 3)² - 9. For y, y² + 4y is (y + 2)² - 4. Substituting gives (x - 3)² - 9 + (y + 2)² - 4 - 12 = 0. Move the three loose numbers across: (x - 3)² + (y + 2)² = 25. So the centre is (3, -2) and the radius is 5. Those two correction terms are the step people forget, and they always move to the right. *(drawing: Each completed bracket leaves a correction behind. Both of them move.)* **Example.** One more question the form answers on sight: is a given point inside, on, or outside? Work out (x - a)² + (y - b)² for that point and compare it with r². Take the circle x² + y² = 36 and the point (7, 1). Then 49 + 1 = 50, which is bigger than 36, so the point sits outside. A smaller value means inside, and an exact match means the point is on the circle itself. No square roots are needed, because comparing the squares gives the same verdict. **Recap.** A circle is every point a fixed distance from a centre, which squares into (x - a)² + (y - b)² = r². Each bracket carries the opposite sign of the centre's coordinate, and the right-hand side is r², not r. An expanded equation gives up its centre only after completing the square in x and in y, with both correction terms moved across. To place a point, compare (x - a)² + (y - b)² with r²: smaller is inside, equal is on, bigger is outside. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Parallel and Perpendicular Lines in Coordinates](https://lightmysky.com/learn/mathematics/parallel-and-perpendicular-lines-in-coordinates-mt_xNEmm139Ka) - [Completing the Square](https://lightmysky.com/learn/mathematics/completing-the-square-mt_rHmCfzB4Fm) ## Opens up - [Lines Meeting Circles: Tangents and Chords](https://lightmysky.com/learn/mathematics/lines-meeting-circles-tangents-and-chords-mt_ELA4IzIVdv)