JentooSAT®

SAT® · Math: Geometry and Trigonometry

The Equation of a Circle in the xy-Plane

The equation (x - h)2 + (y - k)2 = r2 describes every point (x, y) that sits exactly distance r from the center (h, k), which is why it draws a circle.

Watch it drawn4 questionsPractice

A circle is just the set of points that are all the same distance from one fixed point, the center. That fixed distance is the radius. Two squared terms added together is what makes that happen: squaring throws away direction, so (x-h)2 and (y-k)2 only care how far x is from h and how far y is from k, not which way. A parabola comes from only one squared term, so distance in one direction grows without being balanced by the other — there's no fixed radius holding it in.

The equation comes straight from the distance formula. The distance between (x, y) and center (h, k) is √((x-h)2 + (y-k)2). Force that distance to always equal r, and square both sides to clear the root: (x-h)2 + (y-k)2 = r2. That's the whole derivation — nothing hidden, just distance squared set equal to radius squared.

Once you can read this form, two worked moves cover almost everything asked: building the equation from a center and radius, and reading the center and radius back out of a given equation. This sits next to lines and triangles in the same geometry group because circles often get combined with tangent lines, chords, or triangles inscribed in them — so reading center and radius quickly off the equation is what unlocks those harder combined problems.

Key terms

Standard form
The equation (x-h)2 + (y-k)2 = r2, which directly shows the center (h, k) and radius r.
Center
The fixed point (h, k) that every point on the circle is the same distance from.
Radius
The constant distance from the center to any point on the circle, found by taking the square root of the number on the equation's right side.
Distance formula
√((x-h)2 + (y-k)2), the formula that measures straight-line distance between two points.

Watch it explained

  1. How it worksHow does the distance formula turn straight into (x - h)2 + (y - k)2 = r2?
  2. Worked exampleFind the equation of the circle centered at (-3, 5) with radius 4.
  3. Worked exampleFind the center and the radius of the circle given by (x - 2)2 + (y + 7)2 = 45.
  4. Common mistakeWhy doesn't the plus sign inside (y + 7)2 mean the center sits 7 units up?

Check yourself

Your turnmedium1 of 3
A circle has its centre at (4, 3). Write an expression, in terms of x and y, for the square of the distance from the point (x, y) to that centre.

End of topic.

Next: Completing the Square to Find Center and Radius →