Imagine you are an ant walking on a circle. You start your journey from the blue point. Your progress is given in percentages - 100% being a full round trip. In the figure, you've made a progress of 40% and reached the orange spot.

enter image description here

If you're told the percent-progress of your journey (such as 40%), how will you calculate your current coordinates?

You know these:

  • Radius = 50
  • Starting coordinates = (50, 0)
  • Y axis is inverted - positive values are on the bottom (as in figure)
  • Circle border has zero thickness
  • $\begingroup$ Should we assume the percentage is always an integer? $\endgroup$ – Nautilus Aug 9 '15 at 21:25
  • $\begingroup$ @Nautilus No, percent could be a fraction too. $\endgroup$ – Renae Lider Aug 9 '15 at 21:28
  • $\begingroup$ There's an obvious solution, but I have a feeling that it won't be accepted as correct. $\endgroup$ – Nautilus Aug 9 '15 at 21:31
  • $\begingroup$ @Nautilus As Clint Eastwood once said "Go ahead, make my day" $\endgroup$ – Renae Lider Aug 9 '15 at 21:35


$50\sin\left(\frac{360p}{100}\right) + 50$


$50 - 50\cos\left(\frac{360p}{100}\right)$


The centre of the circle is at $(50,-50)$ and its radius is $50$, so the general point on the circle is $$(50+50\cos\theta,50-50\sin\theta)$$ where $\theta$ is the polar angle, i.e. the angle to our current position measured counter-clockwise from the point $(100,50)$.

We want our 'zero' position, or starting position, to be at the top of the circle and our direction to be clockwise, so put $\phi=\frac{\pi}{2}-\theta$. Now $\phi$ is the angle to our current position measured clockwise from the point $(50,0)$. So $$(x_n,y_n)=(50+50\cos\theta,50-50\sin\theta)=(50+50\sin\phi,50-50\cos\phi)$$ and this point is $\frac{\phi}{2\pi}$ percent of the way around the circle.

So the answer is: if you're $N\%$ of the way round the circle, then your position is

$(50+50\sin\frac{2\pi N}{100},50-50\cos\frac{2\pi N}{100})$

(where we take all angles in radians).

  • $\begingroup$ Are you sure, shouldn't there be a 100 in there somewhere? $\endgroup$ – Renae Lider Aug 9 '15 at 21:57
  • $\begingroup$ Since the y axis is inverted, shouldn't the center be at (50, 50)? $\endgroup$ – Nautilus Aug 9 '15 at 22:00
  • $\begingroup$ @Nautilus Good point; I missed that. I'll edit. $\endgroup$ – Rand al'Thor Aug 9 '15 at 22:03
  • $\begingroup$ @RenaeLider Should there? The centre's at (50,50) and we move out from there - where would 100 come from? $\endgroup$ – Rand al'Thor Aug 9 '15 at 22:11
  • $\begingroup$ @RenaeLider Oh, you mean inside the brackets? Another good point - edited once more. $\endgroup$ – Rand al'Thor Aug 9 '15 at 22:13

Not the answer you're looking for? Browse other questions tagged or ask your own question.