Why is the circumference of a circle drawn on a sphere less than the circumference of the circle of same radius drawn on a flat surface?

Question

Why is the circumference of a circle drawn on a sphere less than the circumference of the circle of same radius drawn on a flat surface? This is an extract from the book "The Elegant Universe" but I couldn't figure out why this actually happens. How can one figure out that circumference is going to decrease or increase if a circle is drawn on different types of curved surfaces?

Answer 1

This is probably a question more suited to the maths SE, but here is an answer anyway.

Think about a sphere, radius $R$, with a circle drawn on it. Here is a very scrappy picture:

So, what we want to do is work out the relation between the circumference of the circle and its 'radius', where by 'radius' we mean the arc drawn on the surface of the sphere.

Well, OK, the circle is traced out by a line from the centre of the sphere to the surface making an angle, $\theta$, with the line to the centre of the circle (see the diagram). We can do some elementary trigonometry to find two things:

• the real radius of the circle is $R\sin\theta$ and the circumference of it is therefore $c = 2\pi R \sin\theta$;
• the length of the arc drawn on the sphere is $R\theta$: call this $r$.

So $r = R\theta$, or in other words $\theta = r/R$. So the circumference of the circle is given in terms of $r$ by

$$c = 2\pi R \sin\left(\frac{r}{R}\right)$$

And this is valid for $\theta\in [0, \pi]$ or $r \in [0, \pi R]$.

Well, $\sin x \le x$, so

$$\begin{align} c &= 2 \pi R \sin\left(\frac{r}{R}\right)\\ &\le 2 \pi r \end{align}$$

In other words the circumference of the circle is always less than it would be in flat space.

---

It's useful to look at a couple of particular cases.

First of all, if $r$ is small, then $\sin\left(r/R\right) \approx r/R$ and so

$$\begin{align} c &\approx 2 \pi R \times r/R\\ &= 2 \pi r \end{align}$$

In other words when the circle is small its circumference is approximately what it would be in flat space.

If $\theta = \pi/2$, on the equator of the sphere, then $r = R\pi/2$ and

$$\begin{align} c &= 2 \pi R \sin\left(\frac{R \pi}{2 R}\right)\\ &= 2 \pi R \end{align}$$

While what we would get in flat space is $2 \pi \times R\pi/2$. So the circumference is 'too small' by a factor of $\pi/2$.

Finally if $\theta = \pi$, then $r = \pi R$ and

$$\begin{align} c &= 2 \pi R \sin\left(\frac{\pi R}{R}\right)\\ &= 2 \pi R \sin \pi\\ &= 0 \end{align}$$

This is because we are now at the south pole.

Answer 2

The circumference is the same in both cases if you don't assume the circle's diameter needs to follow the surface of the sphere. However, if you draw the circle's diameter on the surface of the sphere, it is curved, so the straight (true) diameter is less than the curved diameter.

Imagine holding a ring. Simply placing it on the sphere certainly doesn't change its true (uncurved) diameter.

Answer 3

If you look at the diagram in tfb's answer, you can see that the flat radius goes from the edge of the circle, into the sphere, straight horizontally to the circle's center. To get to the circle's center on the sphere, you have to travel the same horizontal distance but you also have to move upward, along the sphere, making the "drawn" radius longer. The circumference is determined by the flat radius which, unless zero, is always shorter than the drawn distance for the reasons above.

Answer 4

I think the crucial ingredient to this is that the difference arises because one assumes that this is from the perspective of "sphere-people" who live confined to the surface of the sphere. They cannot see, hear or reach points interior or exterior to the sphere. The surface of the sphere is their whole world and existence. When these "sphere people" draw a circle, and then walk from the edge to the center of the circle, they don't travel in what we (the "3D" people) would observe to be a straight line. They travel along a curve (the surface of the sphere). So, they (the "sphere-people") measure the radius of their circle to be longer than we would measure it to be. This is because what they see as a "straight line", we see as a curved line, and what we see as a "straight line", they can't see at all.