# ArcLength of Circle: Is this a bug or my error?

Bug introduced in 12.3 or earlier. Fixed in 13.1

In Mathematica 12.3 (Mac) the arc length of a circle, when specified in the form of an ellipse, is incorrect in at least one instance:

Two circles, one specified by only its radius (r=1) and the other as an ellipse (a=r=1,b=r-1):

circles = {Circle[{0,0}, 1, {0, 2 π}],
Circle[{0,0}, {1,1}, {0, 2 π}]}


specify the same circle:

Graphics[{#}, Axes->True]& /@ circles


but the arc lengths differ:

ArcLength /@ circles


{2 π, 4 π}

This occurs when a circle is specified in ellipse (a,b) form and the arc is explicitly given as 2π. Specifying the circle using only the radius or specifying an arc not equal to 2π gives the correct result (values less than 2π give the length of the arc specified; values greater than 2π give the length of the full circle).

I suppose using 2π for any values of arc that are greater than 2π is intentional though it is not specified in the documentation. I'd prefer that asking for an arc greater than 2π would include the length of any repeated section just as ArcLength includes the length of any segments that retrace other parts of a line, such as how:

ArcLength[Line[{ {0,0}, {0,1}, {1,1}, {1,0}, {0,0}, {0,1/2}, {1/2,1/2}, {1/2,0}, {0, 0} }]]


results in 6 rather than 5, even though the last two line segments retrace earlier parts of the line.

• ArcLength[Circle[{0, 0}, {1, 1}, {0, 2 π}]] giving the wrong answer is a bug; please report this to Support. Commented May 30, 2022 at 9:25
• With that said: our standard practice here is to not put in the bugs tag in questions until someone else has confirmed it. Please keep this in mind the next time you ask a question. Commented May 30, 2022 at 9:26
• Sorry about using the bug tag but it was so obviously wrong I couldn't help myself. But I've been wrong at times so I'll avoid doing that in the future. Thanks for confirming the issue. Commented May 30, 2022 at 9:39
• Issue persists in "13.0.0 for Microsoft Windows (64-bit) (December 3, 2021)". Funnily enough, 2.001 Pi or 1.999 Pi for arc angles deliver accurate results. Commented May 30, 2022 at 10:01
• ArcLength[Circle[{0, 0}, {a, b}, {0, 2 π}]] is wrong, too. Commented May 30, 2022 at 20:14