# How is a Circle Just a Bunch of Triangles?

I am somehow supposed to draw how a circle is a bunch of triangles, but I don't see how it is possible.

Can someone help me, and give a few ideas?

• Hint: You're drawing more than just the circle this way. A lot of radial segments are involved. – MackTuesday May 26 '14 at 1:48
• A circle is not a bunch of triangles, so that's not possible. Perhaps you were supposed to approximate it with a lot of really thin triangles? – user2357112 supports Monica May 26 '14 at 1:49
• Are you referencing that the equation for a circle, $x^2+y^2=r^2$, is really just the Pythagorean theorem (which is, of course, about triangles). If so, the reason for that is that a circle is the set of all points that are the same distance from the center and the Pythagorean theorem is how we measure distance. HTH. – Jeff May 26 '14 at 1:52
• Everyone so far is conflating "circle" with "disc". – MackTuesday May 26 '14 at 1:57
• If that's what your teacher said, it's very vague. S/he may have been thinking of approximating a sphere by triangles, maybe? Like in this picture: opengl.org.ru/docs/pg/OpenGL%20Programming%20Guide%202.files/… – bradhd May 26 '14 at 2:05

We might think of a circle as a "bunch" of right-angled triangles with the hypotenuse originating at the center. Alternatively, we can approximate a circle, this can be done e.g. as in the following drawing: The second picture in Rebecca's answer shows that a circle can be approximated as closely as one might wish by a regular polygon with a large enough number of sides.

Each of those triangles has area 1/2 times base times height, and the height is approximately the radius of the circle, and the sum of the bases is the circumference of the polygon. Therefore the area of the polygon is 1/2 times that radial height times the circumference.

It may be tempting to think that this shows only that the area of a circle is approximately 1/2 times radius times circumference.

But just supposing the area of the circle differs from 1/2 times radius times circumference by some unimaginably tiny but nonzero amount. Then one can make the difference less than that tiny amount by choosing a similarly unimaginably large number of sides of the polygon. Conclusion: the area of the circle is exactly 1/2 times circumference times radius.

People like Leibniz and Euler would say simply that there are infinitely many infinitely thin triangles, and the areas of the circle is the sum of their areas.

• Thanks, this really helps me with understanding the picture better. – IHeartBunnies May 26 '14 at 2:30

In addition to Rebecca's answer, you can also approximate the circle by starting with an inscribed equiliteral triangle and drawing more trangles on the edges: You could keep adding triangles on the edges of the previous ones to get very close to a circle

This is common in computer graphics to break down shapes into triangles. Computers like triangles. The disadvantage is that triangles have straight edges so it takes a lot of them to make the ILLUSION of a curve. A circle isn't really a bunch of triangles but it can be very nearly a bunch of triangles. Calculus is based around a similar idea

• I once read a book called The Calculus Direct: An intuitively Obvious Approach to a Basic Understanding of the Calculus for the Casual Observer by John Weiss. I suggest reading it as I learnt the intuition behind calculus. From memory it isn't too expensive – Mmm May 26 '14 at 10:38

Here are 50 triangles that are rotated around the vertex: And here are 50 triangles rotated around the center: 