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lvl10720-Turquoise
20-Turquoise
Dec 7, 2025 4:36 PM UTC
Solved

Area of Parametric (closed) Curve ?


Hello Everyone.
From :

Parametric (closed) Curve.PNG

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  How to calculate Area of Parametric (closed) Curve, the above ? ( MathCad 15 )
  Thanks in advance.
  Regards.
  Loi

DJFAnswer
16-Pearl
Dec 8, 2025 2:16 PM UTC

While this problem is solvable with calculus - there are tricks to doing it other ways that can come in handy for other less mathematical shapes.  Shoelace formula (Surveyor's formula) works just using coordinate points.  Something Werner taught me years ago.

shoelace.png

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14 replies

NickKemaev12-Amethyst
12-Amethyst
Dec 7, 2025 5:22 PM UTC

---------------------------------Screenshot_1.png

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NickKemaev12-Amethyst
12-Amethyst
Dec 7, 2025 6:22 PM UTC

I'm getting better.Screenshot_3.png

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Werner_E
Werner_E25-Diamond I
25-Diamond I
Dec 7, 2025 9:06 PM UTC

Attachments don't work OK at the time, we just see "Virus scan in progress..." and can't download the attachment.
I already notified @unknown  and @PTCModerator  yesterday but so far there was no reaction - its weekend 😞

 

Your curve may look like a cardioide on first glance, but it isn't

Werner_E_0-1765141031559.png

Actually what you show is part of a larger curve with the parameter representation

Werner_E_1-1765141293724.png

and you get the curve you show for t running from 0 to pi

ani1.gif

 

To calculate the area enclosed you can use Green's theorem  which possibly should be attributed to C.F.Gauß.

Werner_E_2-1765141397336.png

Using the parameter representation shown above and the limits 0 and pi for the parameter t, we get the enclosed area with

Werner_E_4-1765141505328.png

ani2.gif

 

 

 

 

 

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    AndrewK
    AndrewKCommunity Manager
    Community Manager
    Dec 8, 2025 12:31 PM UTC

    Hi @Werner_E Thanks for pointing out this issue we are investigating now.

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      Werner_E
      Werner_E25-Diamond I
      25-Diamond I
      Dec 9, 2025 11:09 AM UTC
      @AndrewK wrote:

      Hi @Werner_E Thanks for pointing out this issue we are investigating now.

      Thanks!

      It looks like the problem has now been solved since a few hours ago.

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      DJF16-PearlAnswer
      16-Pearl
      Dec 8, 2025 2:16 PM UTC

      While this problem is solvable with calculus - there are tricks to doing it other ways that can come in handy for other less mathematical shapes.  Shoelace formula (Surveyor's formula) works just using coordinate points.  Something Werner taught me years ago.

      shoelace.png

      Translate
        Werner_E
        Werner_E25-Diamond I
        25-Diamond I
        Dec 8, 2025 3:11 PM UTC

        Yes, and in fact, the shoelace formula can also be traced back to C. F. Gauß! -> Shoelace formula - Wikipedia

        He really was simply a genius.

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