surface area of sphere in a square cylinder are equal

Explanation:
The surface area of a sphere is the same as the lateral surface area of a cylinder with the same radius and a height of 2r. The area of this cylinder is 2πrh = 2πr(2r) = 4πr 2 . To see why the areas are the same, first imagine the sphere sitting snugly inside the cylinder. Then imagine cutting both the sphere and cylinder at any height with two closely spaced planes which are parallel to the bases of the cylinder. Between the planes is a ring shaped strip of the cylinder's surface and a ring shaped strip of the sphere's surface. Both strips have the same area! Why? The strip of the sphere has a smaller radius but a larger width than the strip of cylinder, and these two factors exactly cancel, if the strips are very narrow. The diagram shows two right triangles. The long side of the big triangle is a radius of the sphere and the long side of the small triangle is tangent to the sphere, so the triangles meet at a right angle. You can see that the triangles are similar. This means the ratio of the radii of the two strips (r/R) is the same as the ratio of the widths of the two strips (d/D). So the areas of the strips are equal (2πRd = 2πrD). Since every strip of the sphere has the same area as the corresponding strip of the cylinder, then the area of the whole sphere is the same as the lateral area of the whole cylinder ME- The triangles are similar b/c marked angles =90-x in both cases hwhere x is the Rr angle=> by the angle, angle theorem 2) the sliced sphere area equals 2pi rD and the sliced cylinder= 2pi R d as u can see from the diagram 3) since D/d=R/r=> rD=Rd and if we put this into 2) equation we prove the surface areas are equal! ie Q.E.D.