Volume of a Sphere

In this Instructable we'll cover several ways to find the volume of a sphere - a locus of points that are equidistant to a fixed center in a 3D space.

What you're going to possibly need:

• A Sphere
• Distance measuring tool (ruler, caliper)
• Calculator/Pen+Paper
• Graduated cylinder + water
• A brain (you're on Instructables.com - you've already got this...I'm not going to stress common sense all that much)

Step 1: The Volume Formula

The volume of a sphere is (4/3)*pi*r³
where:

• 4/3 is a constant
• Pi is a constant that for our purposes will = 3.14
• r is the radius of the sphere, which is the distance from the center of the sphere to any point on that sphere's surface

Step 2: Gathering Information

Luckily, the only piece of information we need is the radius of the sphere. There are several ways we can find the radius of a sphere. The more practical ways include deriving the radius from the diameter or the circumference.

You already know this, but:

• Diameter - the longest segment that can exist within the sphere; this is 2*the radius
• Circumference - the length of a circle projected by the sphere; this is 2*pi* the radius

You can quickly approximate the diameter of the sphere by holding a ruler against the longest part of the sphere. Match up the beginning of the ruler with your right eye closed; then take the measurement with your left eye closed. This gives you a terrible approximation, but an approximation nonetheless. After finding the diameter, simply divide it by 2 to yield the radius.

You can find the circumference most easily by taking a string or a wire and wrapping it across the longest portion of the sphere. Then measure the length of the string. After finding the circumference, divide by 2*pi to yield the radius.

Step 3: Plug'n'Play

Once you've found the radius, it's only a matter of plugging it into the equation!

Step 4: Finished...but There's More! =O

Essentially, with a bit of common sense, it only takes the former 2 steps to find the volume of a sphere.

However, maybe some of you wonder why the volume formula is the way it is.

Step 5: Proving the Formula - Disc Integration Method

One way to think of a sphere, is to imagine it as an infinite number of discs. If we found the volume of all of these discs and summed up their individual volumes, we would end up with the volume of the sphere!

With some calculus and some knowledge on the volume of cylinders, we can prove the volume of a sphere!

The pictures below should be able to guide your through the proof...

I've also included the Google SketchUp model, if anyone wants.

Step 6: Proving the Formula - Cylindrical Shell Integration Method

Another way to think of a sphere, is to imagine it as an infinite number of cylindrical shells within one another. The only difference between this and the disc method, is that we're filling the sphere up with a difference shape. But we're still going to sum up the total volume of these shells to find the total volume of the sphere.

The easiest way to find the are of a cylindrical shell is to unravel it into a rectangular prism.

If we recall that the circumference of a circle is 2*pi*radius, and in this case the radius = x , then the circumference will be 2*pi*x, which is the length of the unraveled shell. Other than that, the height of the unraveled shell is still 2y.

Again, the proof is the last picture below. and the SketchUp file is also included

Step 7: Water Displacement Method

A method of finding the volume of a sphere with minimal calculations is to use the Water Displacement Method:

• Fill a beaker or graduated cylinder with enough water to completely immerse the sphere in.
• Record the baseline initial measurement
• Drop the sphere in
• Record final measurement
• Subtract the initial volume from the final volume ~ this is the volume of the sphere!

Step 8: The End

I hope my first Instructable has been helpful in some way. Take care and have fun with this =D.