Equation of a Circle
In short: The equation of a circle, (x − h)² + (y − k)² = r², lists every grid point sitting exactly r units from a fixed center (h, k).
GeometryGeometryG-GPE.A.1
What is Equation of a Circle?
The equation of a circle hides the distance formula inside it. A grid point \((x, y)\) belongs to the curve when its distance to the center \((h, k)\) equals the radius \(r\); squaring both sides erases the root and leaves \((x – h)^2 + (y – k)^2 = r^2\).
Read the signs inside the brackets backward to get the center, and remember the right side holds the radius squared, not the radius. For a center at the origin the pattern shrinks to \(x^2 + y^2 = r^2\).
How to spot it
- An equation has two squared brackets added together and set equal to a positive number: the equation of a circle.
- A problem gives a center and a radius, or a center and one point on the curve, and wants the formula.
- You must read off the center and radius from something like \((x + 4)^2 + (y – 1)^2 = 49\).
- A question asks whether a given point is on, inside or outside a circle.
The method
- Identify the center \((h, k)\): flip the sign of the number next to each variable inside the brackets.
- Find \(r^2\): square the radius, or compute the squared distance from the center to the given point.
- Write \((x – h)^2 + (y – k)^2 = r^2\), taking care with minus signs when \(h\) or \(k\) is negative.
- Test a point: substitute it, and the left side must equal \(r^2\) exactly.
Equation of a Circle: solved example
Problem
A circular fountain on a town map has its center at \((3, -2)\) and a radius of 5 units. Write the equation and check whether the bench at \((7, 1)\) sits on the rim.
Solution
With \(h = 3\), \(k = -2\) and \(r = 5\): \((x – 3)^2 + (y + 2)^2 = 25\).
Substitute \((7, 1)\): \((7 – 3)^2 + (1 + 2)^2 = 16 + 9 = 25\). The left side matches the right side, so the bench is on the rim.
Try it
Question 1
Find the center and radius of the circle \((x + 4)^2 + (y – 1)^2 = 49\).
Show the answer
The center is \((-4, 1)\) and the radius is \(\sqrt{49} = 7\).
Question 2
Write the equation of the circle centered at the origin that passes through \((6, 8)\).
Show the answer
\(r^2 = 6^2 + 8^2 = 100\), so the equation is \(x^2 + y^2 = 100\).
Equation of a Circle vs. Distance Formula
The distance formula returns one number, the gap between two known points, and keeps its square root. A circle’s equation is a rule many points obey, built from that gap squared. The first answers how far apart; the second answers which points are exactly that far from the middle.
Keep going
Lessons
Worksheets and quizzes
Related words
- Distance Formula
- Radius and Diameter
- Pythagorean Theorem
- Completing the Square
- Parabola
- Conic Sections
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