Standard form of a circle equation — center and radius read directly from the equation.
The Mnemonic
Every Point on a Circle, One Equation
The standard equation of a circle with center (h, k) and radius r is: (x − h)² + (y − k)² = r². This equation describes EVERY point (x, y) that lies exactly r units away from the center (h, k) — which is exactly the definition of a circle.
Reading center and radius directly from this form is straightforward once you know the pattern: the numbers subtracted from x and y (with signs flipped) give the center, and the square root of the right side gives the radius.
(x−h)² + (y−k)² = r² — center at (h,k), radius r
💡 Memory Trick
"Standard form of a circle equation — center and radius from the equation." (x−h)² + (y−k)² = r². The center is (h, k) — but watch the signs: if you see (x−3)², the center's x-coordinate is POSITIVE 3, and if you see (x+3)² [= (x−(−3))²], the center's x-coordinate is NEGATIVE 3. The radius is the square root of whatever number sits on the right side.
Why It Works
Straight From the Distance Formula
This equation isn't a separate idea from the Distance Formula — it IS the Distance Formula, just rearranged. A circle is defined as every point that's exactly distance r from a center point (h, k). Plugging (h,k) and (x,y) into the Distance Formula gives: r = √[(x−h)² + (y−k)²].
Squaring both sides to eliminate the square root produces exactly: r² = (x−h)² + (y−k)² — the standard circle equation. Every point (x,y) satisfying this equation is, by construction, exactly r units from the center — nothing more than the Distance Formula applied to the definition of a circle.
Using It In A Proof
Reading and Writing Circle Equations
The two core skills are extracting center/radius from a given equation, and building an equation from a given center and radius.
1
To read center and radius from an equation
Flip the sign inside each parenthesis to get the center's coordinate, and take the square root of the right side to get the radius.
2
To write an equation from a given center and radius
Substitute h and k directly (keeping their original signs, which will appear as subtracted in the equation), and square the radius for the right side.
3
Watch for a center at the origin
If the center is (0,0), the equation simplifies to just x² + y² = r², since (x−0)² = x² and (y−0)² = y².
Full Worked Example
Writing an Equation From a Center and a Point on the Circle
Given: A circle has center (2, −1) and passes through the point (7, 11). Find: the circle's equation.
1
Find the radius using the Distance Formula
The radius is the distance from the center to any point on the circle: r = √[(7−2)² + (11−(−1))²] = √[5² + 12²] = √[25 + 144] = √169 = 13.
2
Substitute the center into the equation
(x − 2)² + (y − (−1))² = r², which simplifies to (x−2)² + (y+1)² = r².
3
Substitute r² (not r)
Since r = 13, r² = 169.
4
Write the final equation
(x−2)² + (y+1)² = 169.
Notice this problem needed the Distance Formula FIRST to find the radius, before the circle equation could even be written — the two topics work together constantly.
🎯 Quick Worked Example
Identify the center and radius of the circle (x+4)² + (y−6)² = 25.
1
Find the center's x-coordinate. (x+4)² = (x−(−4))², so the center's x-coordinate is −4.
2
Find the center's y-coordinate. (y−6)² directly gives a y-coordinate of positive 6.
3
Find the radius. r² = 25, so r = √25 = 5. Center: (−4, 6), Radius: 5.
📌 Exam Application
Watch specifically for a '+' sign inside the parentheses, like (x+4)² — this represents a NEGATIVE center coordinate, not a positive one, since the equation's actual pattern is always (x−h)². Students who don't flip the sign consistently get the center's sign backward.
⚠️ Most Common Equation of a Circle Mistakes
Trap 1 — Forgetting to flip the sign for the center: (x−h)² means the center's coordinate is h with its ORIGINAL sign — but if the equation shows (x+3)², that's actually (x−(−3))², so the center's x-coordinate is −3, not +3.
Trap 2 — Using r instead of r² on the right side: The right side of the equation is always the radius SQUARED, not the radius itself — if a circle has radius 6, the equation's right side is 36, not 6.
✓ Quick Self-Test
1) State the standard equation of a circle. 2) How is this equation derived from the Distance Formula? 3) Identify the center and radius of (x−5)² + (y+2)² = 49. 4) Write the equation of a circle with center (0,0) and radius 8. 5) A circle has center (1,3) and passes through (4,7) — find its equation.