Is a Shape With 1000 Sides the Same as a Circle?

I recently found out about a shape I’d never heard of before: a chiliagon.

A chiliagon is a polygon (a shape made of straight edges) with 1000 sides. In a regular chiliagon, where all the sides are the same length and all the angles are the same size, the angle between two sides is roughly 179.64$^\circ$, so two adjacent sides are very close to being a straight line.

The image below shows a straight line, and an angle of 179.64$^\circ$ sitting on top of it. One side of the angle lies along the straight line, the other doesn’t. Can you tell the difference?

Angle of 179.64 degrees. It looks like a straight line.

What if the lines are longer?

Angle of 179.64 degrees, made from longer lines. You can see that it isn't a straight line.

Now there’s a clear gap on the left-hand side. The further away from the point of the angle you travel, the bigger the gap will get. So just because two things look the same from a particular viewpoint, it doesn’t mean that they are the same.

The next image shows a regular chiliagon – a shape made up of 1000 straight lines. Judge for yourself: is it the same as a circle? If so, why? If not, why not?

Shape made from 1000 straight sides. It looks like a circle.

There’s another way of looking at the chiliagon that might affect your answer. In the next image I’ve added lines from the centre point of the chiliagon to each of the corners. There are 1000 of them.

Shape made from 1000 straight sides, with lines going from the centre to each corner. The outline looks like a circle, but the fill is not solid. It is much darker near the centre than it it near the edges, and there are strange patterns caused by the closeness of the lines.

Does this change your answer? What would it look like if we carried out this process on a circle? Is it even possible to do that?

When you looked at the last image, you might have thought that it doesn’t look like the chiliagon is filled with straight lines. Maybe you can see strange curves and patterns. These are interference patterns caused by the lines being so close together – it’s called a moiré effect. You can play with it in the Geogebra applet below by dragging the slider to change the number of sides. How do the moiré patterns change as you change the number of sides? When does the shape stop looking like a circle?

Have fun!





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Hello! I’m Sam Hartburn, a freelance maths author, editor and animator. I also dabble in music and write mathematical songs. Get in touch by emailing or using any of the social media buttons above.

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