Circle Equation Calculator
Completing the square turns the general equation into center–radius form: x² + y² − 4x + 6y − 12 = 0 has center (2, -3), r = 5.
center (2, -3), r = 5
- Standard form
- (x − 2)² + (y + 3)² = 25
- General form
- x² + y² − 4x + 6y − 12 = 0
- Radius (decimal)
- 5
- Area πr²
- 78.53981634
- Circumference 2πr
- 31.41592654
Step-by-step solution
- Complete the square: h = −D/2 = 2, k = −E/2 = -3
- r² = h² + k² − F = 25
- (x − 2)² + (y + 3)² = 25
Circle Equation Formula
(x − h)² + (y − k)² = r²
from x² + y² + Dx + Ey + F = 0: h = −D/2, k = −E/2, r² = h² + k² − F
Worked Example
The circle x² + y² − 4x + 6y − 12 = 0:
- Complete the square: h = −D/2 = 2, k = −E/2 = -3
- r² = h² + k² − F = 25
- (x − 2)² + (y + 3)² = 25
General form: x² + y² − 4x + 6y − 12 = 0 · Radius (decimal): 5 · Area πr²: 78.53981634 · Circumference 2πr: 31.41592654
Test-day tip
The SAT often asks for the radius of a circle given in general form. Halve and negate D and E for the center, then r² = h² + k² − F, a 30-second calculation with this pattern.
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Example: The points (1, 2) and (7, 10): Distance d = 10.
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Midpoint Calculator
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
endpoint: x₂ = 2Mx − x₁, y₂ = 2My − y₁
Example: The segment from (−2, 3) to (6, −5): Midpoint = (2, -1).
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Arc Length Calculator
s = rθ, A = ½r²θ
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Example: A circle of radius 6 with a 120° central angle: Arc length s = 12.56637061.
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Special Right Triangles Calculator
30-60-90: 1 : √3 : 2; 45-45-90: 1 : 1 : √2
short leg : long leg : hypotenuse; leg : leg : hypotenuse
Example: A 30-60-90 triangle with short leg 5: Sides: short leg = 5, long leg = 5√3, hypotenuse = 10.
Frequently asked questions
How do I complete the square for a circle?
Group x and y terms, add (D/2)² and (E/2)² to both sides, then write each group as a squared binomial.
How do I tell if a point is inside a circle?
Substitute it into (x − h)² + (y − k)²: less than r² means inside, equal means on the circle, more means outside.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Circle Equation Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/circle-equation-calculator/
IEEE
Scientific Converters, "Circle Equation Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/circle-equation-calculator/