Surface area of spheres

A solid 3-dimensional circular object is a sphere.  Its surface is defined as the collection of points that are all equidistant from a central point (centre of the sphere).  Half of a sphere is called a hemisphere.

Unlike solids we have seen so far, we cannot unwrap a sphere, get a 2D net and calculate its area.  So the surface area of a sphere is approached in a different way.  Archimedes discovered that a cylinder that circumscribes a sphere, as shown in this picture, has a curved surface area equal to the surface area, S, of the sphere. 

Archimedes showed that the surface area of the circular component of the cylinder wrapping the circle, has area $2\pi r\times2r=4\pi r^2$2πr×2r=4πr2 and that this area is the same value for the surface area of a sphere.  

Here is a detailed explanation of the proof

In what follows, we will prove that the surface area of a sphere is the same as the area of the curved surface of a cylinder with the same radius and a height of $2r$2r, namely $2\pi rh=2\pi r\times2r$2πrh=2πr×2r = $4\pi r^2$4πr2 . 

To see why the areas are the same, first imagine the sphere sitting neatly inside the cylinder. Then imagine cutting both the sphere and cylinder at any height with two closely spaced sections, parallel to the bases of the cylinder. Between these sections is a ring shaped strip of the cylinder's surface and a ring shaped strip of the sphere's surface. The formula for the surface area of a sphere says that these strips have the same area, let's have a look at why.  

The strip of the sphere has a smaller radius but a larger width than the strip of cylinder, and if the strips are narrow, these two factors exactly cancel. The diagram above also 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 ($\frac{r}{R}$rR​) is the same as the ratio of the widths of the two strips ($\frac{d}{D}$dD​).

That is $\frac{d}{D}=\frac{r}{R}$dD​=rR​ and $Rd=rD$Rd=rD.

So the areas of the strips are equal ($2\pi Rd=2\pi rD$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 curved area of the whole cylinder, namely $4\pi r^2$4πr2 .

Worked Examples

Question 1

Question 2

Question 3