The altitude of a zone of one base is 4cm, the radius of the base is 2 cm. Find the area of the sphere?
The sphere has centre O and radius . You'll see that the 'base' is (I think) a plane that cuts the sphere, forming a circle, centre B. The radius of the circle of this cut, BC, is 2 cm. The line joining B to O is then produced to meet the sphere at A. It's this line that I think is the altitude of the zone. So BA = 4 cm.
(I've always thought a zone was part of the surface of a sphere that lies between two parallel planes, and that what is referred to in this question is called a spherical cap. But there may be other meanings that I don't know about!)
So, if this is the correct diagram, the working is fairly easy. Use Pythagoras' Theorem on the triangle OBC:
You'll find if you simplify this equation, you get a very easy value for . Then you can use the formula for the surface area of a sphere:
to complete the solution.