The spindle and the wedge: Surface area and volume

The spindle is the exterior surface of the volume that has the shape of a segment of a mandarine or orange, which is known as a wedge.

The time zones are (although the Earth is not exactly spherical) the most habitual example of this figure.

If we apply proportions according to the area of the sphere (with the degree of opening of the spindle $n$) we find that the area of the spherical spindle is: $$A_{spindle}=4\pi \cdot r^2 \cdot \dfrac{n_{degrees}}{360^\circ}$$

Using the same procedure but with the expression of volume, we can see that the volume of the spherical wedge is: $$V_{wedge}=\dfrac{4}{3}\pi \cdot r^3 \dfrac{n_{degrees}}{360^\circ}$$