Take a compass and draw a circle on a globe. Then take the same compass, opened to the same width, and draw a circle on a flat piece of paper. Which circle has more area?

If the circle is small compared to the radius of the globe, then the two circles will be approximately equal because a small area on a globe is approximately flat.

To get an idea what happens for larger circles, let’s a circle on the globe as large as possible, i.e. the equator. If the globe has radius *r*, then to draw the equator we need our compass to be opened a width of √2 *r*, the distance from the north pole to the equator along a straight line cutting through the globe.

The area of a hemisphere is 2π*r*². If we take our compass and draw a circle of radius √2 *r* on a flat surface we also get an area of 2π*r*². And by continuity we should expect that if we draw a circle that is nearly as big as the equator then the corresponding circle on a flat surface should have approximately the same area.

Interesting. This says that our compass will draw a circle with the same area whether on a globe or on a flat surface, at least approximately, if the width of the compass sufficiently small or sufficiently large. In fact, we get *exactly* the same area, regardless of how wide the compass is opened up. We haven’t proven this, only given a plausibility argument, but you can find a proof in [1].

Note that the width *w* of the compass is the radius of the circle drawn on a flat surface, but it is **not** the radius of the circle drawn on the globe. The width *w* is greater than the radius of the circle, but less than the distance *along the sphere* from the center of the circle. In the case of the equator, the radius of the circle is *r*, the width of the compass is √2 *r* , and the distance along the sphere from the north pole to the equator is *πr*/2.

## Related posts

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[1] Nick Lord. On an alternative formula for the area of a spherical cap. The Mathematical Gazette, Vol. 102, No. 554 (July 2018), pp. 314–316

This broke something in my brain – so will put this out here and wait for community illumination —

Slice a sphere of radius R in half, (along equator, say) and place it on a table, the area of the planar circle is clearly πR^2. The area of the hemisphere is half that of the full sphere (4πR^2) …so 2πR^2.

The area of the hemisphere and the planar circle which it projects down to are clearly unequal.

I think I understand the source of the confusion, and so I added one more paragraph to the post to explain. The key is that the compass width is

notthe radius of the circle on the sphere.Illumination Unlocked. Thanks John.