How this instrument works
A circle's standard-form equation, (x−h)²+(y−k)²=r², shows its center (h,k) and radius r directly. Expanding both squared terms and collecting everything on one side produces the general form, x²+y²+Dx+Ey+F=0, with three shortcut formulas connecting the two: D=−2h, E=−2k, and F=h²+k²−r² — reached by literally expanding (x−h)² and (y−k)² and gathering like terms, with no further algebra needed once the pattern is known.
This is the exact reverse of this site's companion General to Standard Form page, which completes the square to go the other direction. Together, the pair let a circle's equation move freely between the compact, center-revealing standard form and the expanded, everything-multiplied-out general form that some contexts — solving a system involving a circle and a line, for instance — actually require.
A quick sanity check any expansion should pass: plugging the center point (h,k) itself back into the resulting general-form equation should NOT satisfy it (the center isn't ON the circle), while any point genuinely on the circle's edge, a distance r from center, will satisfy it exactly.
- Enter the circle's center coordinates into the Center: x (h) and Center: y (k) fields.
- Enter the circle's radius into the Radius field.
- Read D, E, and F: the sheet expands the standard-form equation directly.
Worked example — center (2,3), radius 3
A circle is centered at (2, 3) with radius 3. Expanding gives D = −2(2) = −4, E = −2(3) = −6, and F = 4+9−9 = 4 — the general-form equation x² + y² − 4x − 6y + 4 = 0, the exact reverse of what the companion General to Standard Form page would recover from these same three coefficients.
A circle centered at the origin with radius 5 has D = 0, E = 0, F = −25 — no linear terms at all, since a circle centered at (0,0) needs none. A circle centered at (3,−4) with radius 5 gives D = −6, E = 8, F = 0 — a 3-4-5 relationship between the center and radius making F come out to exactly zero.
Questions
How do you expand a circle's standard form into general form?
Multiply out (x−h)² and (y−k)² and collect the resulting terms — the shortcuts D=−2h, E=−2k, and F=h²+k²−r² give the same result directly, without writing out every intermediate expansion step by hand.
How is this different from the General to Standard Form page?
That page starts from general form and completes the square to recover the center and radius. This page runs the opposite direction — starting from a known center and radius, and expanding out to the general-form coefficients instead.
What does F represent geometrically?
F = h²+k²−r² is what remains once both squared terms are expanded and the r² is moved to the same side as everything else — it isn't a standalone geometric length on its own, but a specific combination of the center's distance from the origin and the circle's radius.
Why might general form be more useful than standard form?
General form is the shape needed when a circle's equation is being combined algebraically with another equation — solving for where a circle and a line intersect, for instance, is more directly set up starting from the fully expanded general form.
What if the circle is centered at the origin?
Both D and E come out to exactly 0, since h and k are both 0 — the general-form equation simplifies to just x² + y² + F = 0, with F equal to negative the radius squared.