Convert between standard form (x−h)²+(y−k)²=r² and general form x²+y²+Dx+Ey+F=0. Find center, radius, area, and circumference from either equation form.
The equation of a circle can be written in two main forms. The standard (center-radius) form is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius. The general form is x² + y² + Dx + Ey + F = 0, where D, E, and F are constants related to the center and radius by D = −2h, E = −2k, and F = h² + k² − r².
Converting between these forms is a common task in coordinate geometry. Going from standard to general form requires expanding the squares and rearranging. Going from general to standard form requires completing the square for both x and y terms. This calculator handles both directions instantly.
Beyond the equation itself, the calculator computes the circle's area (πr²), circumference (2πr), and checks whether the circle passes through the origin. It also shows the distance from the origin to the center, which is useful for understanding the circle's position in the coordinate plane. Whether you're a student working through conic sections, a teacher preparing examples, or an engineer defining circular boundaries, this tool converts and analyzes circle equations in one step. Use the presets to explore classic examples like the unit circle or circles passing through the origin.
This circle equation calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Conversion Direction, Center x-coordinate (h), Center y-coordinate (k), Radius (r) and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Standard form: (x − h)² + (y − k)² = r² General form: x² + y² + Dx + Ey + F = 0 Relationships: D = −2h, E = −2k, F = h² + k² − r² Center: (−D/2, −E/2) Radius: r = √(h² + k² − F) = √((D/2)² + (E/2)² − F) Area: A = πr² Circumference: C = 2πr
Result: For form=standard, h=0, k=0, the tool returns the solved circle equation outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in circle equation formulas and reports derived values, checks, and classifications automatically.
This page is tailored to circle equation, with outputs tied directly to the form fields (Conversion Direction, Center x-coordinate (h), Center y-coordinate (k), Radius (r)). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
The standard form is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius. It directly shows the center and radius.
The general form is x² + y² + Dx + Ey + F = 0. To find the center and radius, you complete the square or use: center = (−D/2, −E/2), r = √((D/2)² + (E/2)² − F).
Complete the square for x and y terms. Group x terms and y terms, add the needed constants to both sides, and factor into perfect squares.
The equation represents no real circle — the radius would be imaginary. This means the equation has no real graph.
Substitute (0, 0) into the equation. In general form, this means F = 0. In standard form, h² + k² = r².
No. This calculator is specifically for circles, where the x² and y² coefficients are both 1. Ellipses have different coefficients for x² and y².