Spherical layer - area and volume

A spherical layer is the solid that arises when a sphere is cut by two parallel planes σ1 and σ2, both of which are less than the radius r away from the centre S of the sphere. The planes cut two circles out of the sphere, with radii ρ1 and ρ2, which form the bases of the layer, and the part of the sphere between them is exactly the spherical layer. The curved lateral surface of this layer is called a spherical zone, and it is the patch of the spherical surface bounded by two parallel circles. If one of the cutting planes only touches the sphere, both bases merge into a single one and instead of a layer we get a spherical segment topped by a spherical cap.

An interesting property of the spherical zone is that its area depends only on the radius of the sphere and on the layer height v, that is on the perpendicular distance between the two planes, and not on where exactly the sphere is cut. Two equally high zones on the same sphere therefore always have the same surface area, even if one of them lies at the equator and the other close to a pole. Spherical layers turn up when calculating the surface area and volume of barrel-shaped tanks, domes, floats or lenses, and in geography when determining the areas of climate belts between two parallels. That makes it not only a school exercise, but also a practical tool in engineering, construction and astronomy.