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.
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, the page converts and analyzes circle equations from the same input set. Use the presets to explore classic examples like the unit circle or circles passing through the origin.
Use this page when you need to move between center-radius form and general form without losing track of the algebra. It is helpful for conic-sections homework, graphing checks, CAD sketches, and coordinate-geometry problems because it ties the equation, center, radius, and derived measurements back to the same set of coefficients.
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 = 2πrResult: 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.
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. Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.
Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.
Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.
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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.
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