The Equation of a Circle: Center, Radius, and the Distance Idea
Master the standard equation of a circle (x − h)² + (y − k)² = r², find the center and radius, and complete the square to convert from general form — with worked SAT and ACT problems.
The Short Version
- Standard form: (x − h)² + (y − k)² = r², with center (h, k) and radius r.
- Mind the signs: (x − 3) means h = +3; the right side is r², so take the square root for r.
- Build the equation from a center and radius, or read them off a given equation.
- Convert from general form by completing the square. An SAT/ACT and Algebra II topic.
A circle is defined by one simple idea: it's the set of all points exactly r units from a center point. Translate that sentence into algebra using the distance formula and you get the standard equation of a circle — an equation in which the center and the radius are visible at a glance, just like vertex form does for parabolas.
This guide derives the standard form, shows how to read and write it, and covers completing the square to convert, with worked and practice problems matched to real test difficulty at Northside Tutoring.
Why the Circle Equation Matters
The circle equation is a recurring SAT topic and appears on the ACT too. Questions ask for the center and radius, whether a point is on the circle, or to convert from a messy expanded form. It's coordinate-geometry content beyond the SSAT, built on distance and completing the square.
Standard Form
The standard equation of a circle with center (h, k) and radius r is:
It's literally the distance formula, squared: every point (x, y) whose distance to (h, k) equals r satisfies it.
Reading It Off the Graph
The center (h, k) and the radius r are exactly the quantities in the standard equation.
Center & Radius From the Equation
Given (x − 2)² + (y + 3)² = 25, read off the center and radius — watching the signs. The center is (2, −3) (the +3 means k = −3), and r = √25 = 5.
Two sign traps
First, the center signs flip: (y + 3) gives k = −3. Second, the right side is r², not r — if it equals 25, the radius is 5, not 25. Both trip up students constantly.
Writing the Equation
To write a circle's equation, drop the center and radius into the form. Center (−1, 4), radius 6: (x + 1)² + (y − 4)² = 36. Remember to square the radius on the right.
Completing the Square
Sometimes the circle is given in expanded (general) form like x² + y² − 6x + 4y − 12 = 0. To find the center and radius, group the x-terms and y-terms and complete the square on each, turning the equation back into standard form. Then read off (h, k) and r as before.
Where You'll See This — Test by Test
Not on a reference sheet — you must know the form. The SAT regularly tests center/radius and completing the square; the ACT tests the standard form. It's beyond the SSAT.
Digital SAT
A recurring topic: identifying center and radius and converting from general form by completing the square.
Explore SAT Tutoring → College AdmissionsACT
Tests reading the center and radius from standard form and writing a circle's equation.
Explore ACT Tutoring → Independent School AdmissionsSSAT
Not on the SSAT — coordinate geometry beyond its scope. Build circle and distance basics with earlier prep first.
Explore SSAT Tutoring → K-12 CurriculumGeometry / Algebra II
A core coordinate-geometry topic in Geometry and Algebra II.
Explore Math Tutoring →Watch the Lesson
Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.
Circle Equations — In Plain English
A live walkthrough from our tutoring team.
— Featuring a Northside Tutoring instructor
Worked Example Problems
These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.
What is the center and radius of (x − 4)² + (y − 1)² = 9?
Show solution
Center (4, 1); r = √9 = 3.
What is the center of (x + 5)² + (y − 2)² = 16?
Show solution
Signs flip on the center: (x + 5) gives h = −5; (y − 2) gives k = 2.
Write the equation of a circle with center (3, −2) and radius 7.
Show solution
(x − 3)² + (y + 2)² = 49 (square the radius).
A circle is (x − 1)² + (y − 1)² = 25. Is the point (4, 5) on the circle?
Show solution
Check: (4 − 1)² + (5 − 1)² = 9 + 16 = 25. Yes — it satisfies the equation.
Find the center and radius: x² + y² − 6x + 4y − 12 = 0.
Show solution
Complete the square: (x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4.
(x − 3)² + (y + 2)² = 25, so center (3, −2), r = 5.
Common Mistakes to Avoid
Three traps that catch students every year
- Forgetting the center signs flip. (x + 5) means h = −5; the equation subtracts h.
- Reading r² as r. The right side is the radius squared — take its square root for the actual radius.
- Botching completing the square. Add the same amount to both sides, and halve-then-square the linear coefficient.
Practice Problems — You Try
Three problems below. Work each before checking the solution.
Center and radius of (x − 2)² + (y + 6)² = 49?
Show solution
Center (2, −6); r = √49 = 7.
Write the equation: center (0, 0), radius 4.
Show solution
x² + y² = 16.
Find the radius: x² + y² + 8x − 10y + 5 = 0.
Show solution
Complete the square: (x + 4)² + (y − 5)² = −5 + 16 + 25 = 36. So r = 6.
The Northside Method — How We Teach This 1-on-1
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