Surface Area of a Sphere

The surface area of a sphere is the quantity of units of length squared that will cover the surface of a sphere. In Dimensional Analysis, this can be written as $L^2$.

Geometrically speaking, the sphere is a unique shape. Out of all possible shapes, the sphere is the shape that minimizes surface area per volume. Meaning the sphere encloses the greatest possible volume with the smallest possible surface area.

This fact, and given that nature at all major scalar resolutions of reality tends to replicate itself with the sphere, will eventually reveal something profound to us about the fundamental nature of reality. In SI Units we generally measure length in meters ($m$), and area in meters squared $m^2$. The surface area of a sphere is also measured in square units.

There are any number of symbols that we can use to denote the surface area of a sphere from $a$, to $SA_s$, $A$, or $A_s$. Whatever the symbol, the formula for the surface area of a sphere is given by

\[ SA_s = 4 \pi r^2 \]

From a known surface area of a sphere we can manipulate the above expression to solve for the radius

\[ r = \sqrt{\frac{a}{4 \pi}} \]

Interesting Aspects of the Surface Area of a Sphere

There are a few interesting pieces that I would like to draw your attention to:

First is that the surface area of a sphere varies exponentially by the radius. As the radius increases the sphere’s surface area increases exponentially.

A simple mathematical analysis will show how quickly the surface area increases as the radius increases. I am going to ignore the $4 \pi$ factor, and just focus on the relationship between radius and surface area. (Though all you have to do to get a precise or estimated value for the surface area from the following numbers is multiply by $4 \pi$, for a precise value, or 12 for an estimated value.)

From the relationship $SA_s \propto r^2$ we can see how quickly the surface area rises with relatively small increases in the sphere’s radius.

\begin{equation*} \begin{align} SA_{7} \propto (7)^2 = 49 \\ SA_{10} \propto (10)^2 = 100 \\ SA_{12} \propto (12)^2 = 144 \\ SA_{15} \propto (15)^2 = 225 \\ SA_{20} \propto (20)^2 = 400 \\ SA_{25} \propto (25)^2 = 625 \end{align} \end{equation*}

Another interesting thing about a sphere’s surface area is it’s relationship to the area of a circle. The formula for the area of a circle is given by $A_{circle} = \pi r^2$. This is fascinating. A sphere’s surface area is exactly 4x that of the area of a circle of the same radius.

When we apply dimensional analysis to this formula, you will notice that aside from two dimensionless quantities ($2$ and $\pi$) the only dimension we have is length. Length squared specifically, because that is the definition of area, giving a sphere’s surface area the dimension $L^2$.

Archimedes Discovery

The Greek mathematician, astronomer, physicist, engineer, and inventor Archimedes discovered that the surface area of a sphere was equal to the lateral surface area of a cylinder almost 2300 years ago.

Archimedes discovered this under the specific condition that the diameter of the sphere in question was equal to the height of the cylinder, and the radius of the sphere was equal to the radius of the cylinder.

As you will know, the radius of a cylinder is the radius of the circle on the top or the bottom of the cylinder, while its height refers to the height or length of the cylinder. The lateral surface area of a cylinder is the surface area of the tube itself, without considering the surface area of the circular top and base of the cylinder, which when stretch out flat (if we cut it down its length) is actually a rectangle.

Lateral is a word the refers to the sides of something. The lateral surface area of a cylinder is equal to

\[ SA_L = 2 \pi r h \]

where $r$ is the radius of the circle at the top or bottom of cylinder, and $h$ is the height (or length) of the cylinder, depending on how you look at it.

This gives us the following relationship between a sphere’s surface area and the lateral surface area of a cylinder, only when the radius of the cylinder equals the radius of the sphere ($r=r$) and the height of the cylinder equals the diameter of the sphere ($h=d=2r$). From this equality we can derive the equation for the surface area of a sphere as follows

\[ SA_s = 2 \pi r h = 2 \pi r (2r) = 4 \pi r (r) = 4 \pi r^2 \]

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