Circle Equations

Equation Setup

(x − 2)² + (y + 3)² = 25

Center h

Center k

Radius r

Enter the center (h, k) and radius r from the standard form equation.

Examples

x² + y² − 4x + 6y − 3 = 0

From x² + y² + Dx + Ey + F = 0. Complete the square to find standard form.

Examples

(x − 2)² + (y + 3)² = 25

Center h

Center k

Radius r

Point x₀

Point y₀

Check whether (x₀, y₀) is inside, on, or outside the circle.

Examples

Result

Enter values above and press Calculate to see the circle properties, converted form, and graph.

Standard Form

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General Form Expansion

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Center

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Radius

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Diameter

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Area

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Circumference

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r²

Standard Form

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Center

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Radius

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r²

Check: Expand Back to General Form

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Result

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Distance Calculation

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Standard Form

(x − h)² + (y − k)² = r²

Center: (h, k) — the point equidistant from every point on the circle.

Radius: r — always positive. r² appears on the right side.

Watch the signs! If the equation is (x + 3)² + (y − 1)² = 16, then h = −3, k = 1, r = 4. The sign inside the parenthesis is the opposite of h and k because (x − h) means (x − (−3)) = (x + 3).

r² must be positive. If you expand and get r² = 0, the "circle" is just a single point. If r² < 0, there is no real circle.

Completing the Square

x² + Dx = (x + D/2)² − (D/2)²

To convert from general form x² + y² + Dx + Ey + F = 0:

1. Group x-terms and y-terms: (x² + Dx) + (y² + Ey) = −F

2. Complete the square for x: add (D/2)² to both sides.

3. Complete the square for y: add (E/2)² to both sides.

4. Result: (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² − F

So h = −D/2, k = −E/2, and r² = (D/2)² + (E/2)² − F.

Always add the same amount to both sides when completing the square.
The half-coefficient trick: take D, divide by 2, then square it.
Check your answer by expanding back to general form.

Circles still confusing?