# Center of a Circle: Definition, Formula, How to Find TL;DR The center of a circle is the single fixed point that sits the same distance, the radius, from every point on the circle. This article covers the definition, the equation form `(x−h)²+(y−k)²=r²` and three ways to find the center: from the equation, from the endpoints of a diameter, and from three points on the circle. ## What Is the Center of a Circle? The **center of a circle** is the fixed point in the plane that is **equidistant from every point on the circle**. That equal distance is the **radius**. A circle is, by definition, the set of all points a fixed distance from one point, and that one point is its center. A few facts follow straight from this definition, no formula needed: - The center is where **all the radii meet**. Draw any radius, draw another, and they cross at the center. - The center is the **midpoint of any diameter**. A diameter is a chord that passes through the center, so the center splits it exactly in half. - The center does **not lie on the circle itself**. It sits inside, at distance `r` from the curve, not on it. In coordinate geometry the center is written as the point `(h,k)`. The whole job of "finding the center" is finding the two numbers `h` and `k`. ## How to Find the Center of a Circle There is no single button for this because the center depends on **what you are given**. Three situations come up again and again, and each has its own clean method. Here are all three, so you know every route exists rather than memorizing just one. ### From the equation of the circle The standard form of a circle's equation is: `(x−h)²+(y−k)²=r²`, where `(h,k)` is the center and `r` is the radius. Read directly: the center is the point you subtract inside each squared bracket. **Watch the signs**, the equation has minus signs built in, so `(x−3)²+(y+2)²=25` has center `(3,−2)`, not `(3,2)`. When the equation arrives in **general form**, `x²+y²+Dx+Ey+F=0`, it is not yet readable. You **complete the square** on the `x` terms and on the `y` terms to fold it back into standard form, and the center appears. ### From the endpoints of a diameter If you know the two ends of a diameter, the center is simply the **midpoint** of that segment, because the center always cuts a diameter in half. With endpoints `(x₁,y₁)` and `(x₂,y₂)`: `(h,k)= \left(\frac{x₁+x₂}{2}, \frac{y₁+y₂}{2}\right)`. ### From three points on the circle Given three points the circle passes through, the center is the one point equidistant from all three. The clean geometric route: the center lies on the **perpendicular bisector** of any chord. Draw the perpendicular bisector of two different chords, and they meet at the center. Algebraically, you can set the distances from `(h,k)` to all three points equal and solve the resulting linear system for `h` and `k`. ## Where the Center of a Circle Shows Up The center is not just an exam construction. It is the anchor point of anything circular, and the moment you need to _place_, _aim_, or _spin_ a circular object, the center is the quantity you actually solve for. - **GPS and surveying.** Locating a position by distance from several towers is the **three-circles problem** — the same "find the point equidistant from given points" computation, run in reverse. - **Engineering and manufacturing.** A drilled hole, a turbine shaft, a gear: every rotating part spins about its center. - **Astronomy and navigation.** Circular approximations and their centers are still how we first model orbits, radar sweeps, and satellite footprints. ## Examples of the Center of a Circle ### Example 1 **Find the center of the circle `(x−4)²+(y−7)²=36`.** The equation is already in standard form, so read the center straight off the brackets. The values subtracted from `x` and `y` are the coordinates of the center. `h=4,k=7.` The center is `(4,7)`, and as a bonus `r²=36`, so the radius is `6`. ### Example 2 **Find the center of the circle `(x+5)²+(y−3)²=49`.** A common first move is to read the center as `(5,3)`, copying the numbers as they appear. Check it against the definition: standard form is `(x−h)²+(y−k)²`, with a **minus** built in. Correctly, `(x+5)` gives `h=−5`, and `(y−3)` gives `k=3`. The center is `(−5,3)`. ### Example 3 **Find the center of the circle `x²+y²−6x+8y−11=0`.** This is general form, so complete the square: `(x²−6x)+(y²+8y)=11.` Add 9 and 16 to both sides: `(x−3)²+(y+4)²=36.` The center is `(3,−4)`, and the radius is `6`. ### Example 4 **The endpoints of a diameter are `A(2,1)` and `B(8,9)`. Find the center.** The center is the midpoint of the diameter: `(h,k)=(2+8/2,1+9/2)=(5,5).` ### Example 5 **A diameter runs from `P(−3,6)` to `Q(5,−2)`. Find the center.** Apply the midpoint formula: `h=(-3+5)/2=1`, `k=(6+(−2))/2=2.` The center is `(1,2)`. ### Example 6 **A circle passes through `A(1,1)`, `B(5,1)`, and `C(5,5)`. Find its center.** Use the perpendicular-bisector idea. The center is where the two bisectors cross: `(h,k)=(3,3)`. ## Key Takeaways - The **center of a circle** is the fixed point equidistant from every point on the circle; that distance is the radius. - In coordinate form, the center is `(h,k)` in the equation `(x−h)²+(y−k)²=r²`. - From an equation in general form, complete the square first, then read the center. - From the endpoints of a diameter, the center is their midpoint. - From three points, the center is the circumcenter, found by intersecting perpendicular bisectors, not by averaging.