So, after reading the latest XKCD comic and it accompanying page on the explainxkcd wiki, I saw a link to this site that claims that a sphere has a surface area of approximately 41000 square degrees using this logic:
- A circle has a circumference of 360 degrees
- The radius of a circle is its circumference divided by 2 pi.
- Therefore, the radius of a circle is about 57 degrees.
- The surface area of a circle is equal to its radius squared times 4 pi.
- Therefore, the surface area of a sphere is about 41000 square degrees.
Is this accurate? Is the surface area of a sphere actually 41000 square degrees, or is this proof not actually correct? Is it the same answer you'd arrive at if you did an integration across the surface area of the sphere?
The whole sphere has a solid angle of about 41,000 square degrees (41,253 for a little more precision), but as the comic itself suggests, the dimensional nomenclature is tenuous. Revised points:
$360^{\circ}$ is not the circumference of a circle, but the angle subtended by a full circle from its center.
Fine
One radian is the angle subtended by the arc of a circle (from its center) having the same length as its radius, which in degrees is $360^\circ/(2\pi)=180/\pi \approx 57^\circ.$
Fine
One square radian is the solid angle subtended by a spherical cap (from the spherical center) having the same area as its radius squared, which in square degrees is $(180/\pi)^2\approx 3283 \text{ deg}^2.$ The solid angle subtended by the whole sphere from its center in square degrees is then $4 \pi (180/\pi)^2\approx 41,253 \text{ deg}^2$.