Circle Equation Calculator

How to use the Circle Equation Calculator

1. 1Enter the centre and radius, or D, E and F.
2. 2Pick the units or options from the menus if the circle equation tool offers them.
3. 3Read the circle equation in the large result box.
4. 4Look at the extra rows for related figures.
5. 5Press Copy result to keep the answer.

When you need the circle equation

Students use it for analytic geometry homework. Designers and programmers use it to place and test circles in drawings and games.

If r² comes out as zero the equation is a single point, and if it is negative no real circle exists. The calculator reports this instead of returning a wrong radius.

How the circle equation is worked out

Standard form is (x − h)² + (y − k)² = r². The general form has D = −2h, E = −2k and F = h² + k² − r². Going back, h = −D ÷ 2, k = −E ÷ 2 and r² = h² + k² − F. A circle with centre (2, 3) and radius 5 is (x − 2)² + (y − 3)² = 25, or x² + y² − 4x − 6y − 12 = 0. The second equation gives back D = −4, E = −6, F = −12 and the same circle.

Every answer on this page comes from that one rule, so you can check any result by hand with a calculator. The figures shown underneath, such as other units, are conversions of the same answer.

• Both directions
• Standard and general form
• Warns when no real circle exists

Mistakes to avoid with circle equation

Watch the signs: the centre (2, 3) gives (x − 2) and (y − 3), not plus signs. The general form must have equal coefficients on x² and y², so divide through first if both are 2.

On phone and computer

On a phone, type into the boxes one after another and read the circle equation underneath; Copy result puts it on the clipboard for a message or a note. On a computer the inputs sit on the left and the answer on the right, so you can change one figure and watch the circle equation move. The circle equation calculator works the same on both.

Checking your circle equation by hand

A good habit is to test the circle equation calculator with a case you already know. A circle with centre (2, 3) and radius 5 is (x − 2)² + (y − 3)² = 25, or x² + y² − 4x − 6y − 12 = 0. The second equation gives back D = −4, E = −6, F = −12 and the same circle.

If your own figures give a surprising answer, repeat the working yourself: Standard form is (x − h)² + (y − k)² = r². The general form has D = −2h, E = −2k and F = h² + k² − r². Going back, h = −D ÷ 2, k = −E ÷ 2 and r² = h² + k² − F. Small differences usually come from rounding, from mixing units, or from a typing slip in one of the inputs. A circle equation is easier to trust when you test its parts. Check radius and area with the Circle Calculator, find the gap between two points with the Distance Calculator, and locate a centre between points with the Midpoint Calculator.

Features

• Signs of the centre are reversed in the equation
• Coefficients of x² and y² must match
• Negative r² has no circle
• Clear messages when the circle equation inputs make no sense
• Copy the result with one tap
• Private: nothing is uploaded

Frequently asked questions