Question

Prove the following famous hat box theorem of Archimedes: The surface area of the portion of the sphere of radius r between two horizontal planes is:

a) 4πr^2
b) 2πr^2
c) πr^2
d) (4/3)πr^3

Answer 1

The correct answer for the hat box theorem of Archimedes regarding the surface area of the portion of the sphere between two horizontal planes is b) 2πr². This is calculated by the lateral surface area of the encompassing cylinder, which is height times the circumference of the sphere's great circle.

Step-by-step explanation:

The famous hat box theorem of Archimedes states that the surface area of the portion of the sphere between two horizontal planes is independent of the size of the segment and depends only on the height of the segment. If we take a sphere of radius r, the surface area of such a segment is given by the lateral surface area of the cylinder that would encompass the sphere, since the height of this cylinder is the same as the diameter of the sphere, which is 2r.

The formula for the lateral surface area of the cylinder is the circumference of the base circle times its height. Since the base circle is the great circle of the sphere, its circumference is 2πr, and the height is 2r, the total surface area is 2πr * 2r = 4πr².

So, for the question asked by the student, the correct answer is: b) 2πr².