Surface Area of a Sphere
by victus_maestro in Workshop > Science
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Surface Area of a Sphere
There comes a time in a person's life when they realize something is missing...and oftentimes, they come to realize that it is because they do not know how to calculate the surface area of a sphere!
Alright, maybe not. Chances are, unless you are in some very specialized field, you will not need this on a daily basis. But you will, however, need it for math classes.
So, without further ado, I will proceed to explain how this fun and mysterious procedure is carried out.
Alright, maybe not. Chances are, unless you are in some very specialized field, you will not need this on a daily basis. But you will, however, need it for math classes.
So, without further ado, I will proceed to explain how this fun and mysterious procedure is carried out.
Deconstructing the Sphere
A sphere is a bit like a super-circle...you can rotate a circle around its center in any direction and it will trace the outline of the sphere. As a result, it should make sense that the formula to find the surface area of a sphere is very closely related to that of the one to find the circumference of a circle.
As is shown in the first picture below, the formula for the circumference of a circle is 2(pi)(r). The second one shows how similar the sphere formula is, being 4(pi)(r)2. Why is the formula in this format? The answer lies in radian measurement.
If you are not familiar with radian measures, you will encounter them in trigonometry at the latest. Many people learn about them in high school Algebra 2, but they are not explained very well and so they seem a bit mysterious.
The next step will be a very basic description of how radian measurement works as relates to these formulas. If you are already familiar with them, or don't want the theory, by all means skip ahead.
As is shown in the first picture below, the formula for the circumference of a circle is 2(pi)(r). The second one shows how similar the sphere formula is, being 4(pi)(r)2. Why is the formula in this format? The answer lies in radian measurement.
If you are not familiar with radian measures, you will encounter them in trigonometry at the latest. Many people learn about them in high school Algebra 2, but they are not explained very well and so they seem a bit mysterious.
The next step will be a very basic description of how radian measurement works as relates to these formulas. If you are already familiar with them, or don't want the theory, by all means skip ahead.
The Radian Relationship
Radians are, at their heart, a function of proportion. They are defined by the relationship of a circle's circumference to it's radius--this is the reason that pi is always 3.14159 and not a changing value. It can be used interchangeably with degree measurement, but is more difficult to calculate because pi almost always plays a part in the calculation.
So if it can be used like degrees, but is harder to use, why use it at all, you ask?
The big difference is that it comes in handy for things like surface area, circumference, area, and volume because, unlike degrees, radians return the same unit of measurement that you measure the radius in. In other words, things boil down to distance, and not angles. Take a look at the picture below.
2(pi)r, the base formula for circumference of a circle, returns 6.28 with a radius of 1. That means 6.28 radians, which essentially translates to 6.28 times the radius. If your radius is 1 inch, your circumference is 6.28 inches. This cannot be done with degrees, and if it could, it would return a unit like 360 degree-inches, and that would make our job very difficult if not downright impossible.
That just about covers what you need to know about radians at this point, I'll just finish up with some useful tidbits:
pi (3.14) radians is half a circle. As would logically follow, 2pi (6.28) is the whole circle, in terms of the radius. That is why 2(pi)(r) is circumference--you are essentially multiplying a whole circle by however long the radius is--the relationship will always be proportional.
Okay, enough theory. Let's get on with how to actually do this thing.
So if it can be used like degrees, but is harder to use, why use it at all, you ask?
The big difference is that it comes in handy for things like surface area, circumference, area, and volume because, unlike degrees, radians return the same unit of measurement that you measure the radius in. In other words, things boil down to distance, and not angles. Take a look at the picture below.
2(pi)r, the base formula for circumference of a circle, returns 6.28 with a radius of 1. That means 6.28 radians, which essentially translates to 6.28 times the radius. If your radius is 1 inch, your circumference is 6.28 inches. This cannot be done with degrees, and if it could, it would return a unit like 360 degree-inches, and that would make our job very difficult if not downright impossible.
That just about covers what you need to know about radians at this point, I'll just finish up with some useful tidbits:
pi (3.14) radians is half a circle. As would logically follow, 2pi (6.28) is the whole circle, in terms of the radius. That is why 2(pi)(r) is circumference--you are essentially multiplying a whole circle by however long the radius is--the relationship will always be proportional.
Okay, enough theory. Let's get on with how to actually do this thing.
Putting the Formula to Use (the LONG Step)
Now that you are all primed on how this works, let's work it out. I already gave the formula in Step 1, but here we'll be able to see it in action. Or rather, see it being calculated. I can't say there's any real action involved.
To make the numbers more interesting, let's come up with a hypothetical situation in which this would be used.
Let's see...an eccentric artist friend of yours (everyone's got at least one, right?) is designing an installation piece for a local art museum. He has designed a series of differently-sized spheres from various materials to represent the diversity of humanity and relate mankind's struggle to the dynamic forces holding together space itself. However, he has found that he wants to paint them all to show a common sociological unity, hiding the fact that they are different underneath. He has enlisted your help to calculate how much paint he needs because he is so tired from making them that he cannot do it himself.
He also wants you to buy the least amount of paint possible, because, as mentioned, he is an artist, and is therefore poor.
There is a 2-foot diameter wooden ball, a 3-foot diameter bronze ball, and a 4-foot diameter concrete ball. How much paint (in terms of square inches covered) would be needed to cover them and have none left over?
This is really deceptively easy. Just plug the values into the formula:
BALL 1 (2 FOOT DIAMETER):
This one is 2 feet across, so we convert to inches (24) and half it (12) to get the radius in inches.
Then we plug in the formula values:
4 (pi) (r)2 (Where r is the radius)
4 (pi) 122
4 (pi) 144
12.566 (144)
1,809.56 square inches
Simple, no? Let's get the same thing going with the second one.
BALL 2 (3 FOOT DIAMETER):
This one is 3 feet, so we cut that in half and convert to inches again (18). And then plug it in:
4 (pi) (r)2
4 (pi) 182
4 (pi) 324
12.566 (324)
4,071.50 square inches
Now, I would walk you through the third one, but I think you get the idea. I will do the quick sum up.
BALL 3:
12.566 (24)2
12.566 (576)
7,238 square inches
So, add those up and you get the amount of square inches you need to cover:
1,809.56 + 4,071.50 + 7,238 = 13,119.06 square inches (or 91 square feet).
Now go buy that paint!
To make the numbers more interesting, let's come up with a hypothetical situation in which this would be used.
Let's see...an eccentric artist friend of yours (everyone's got at least one, right?) is designing an installation piece for a local art museum. He has designed a series of differently-sized spheres from various materials to represent the diversity of humanity and relate mankind's struggle to the dynamic forces holding together space itself. However, he has found that he wants to paint them all to show a common sociological unity, hiding the fact that they are different underneath. He has enlisted your help to calculate how much paint he needs because he is so tired from making them that he cannot do it himself.
He also wants you to buy the least amount of paint possible, because, as mentioned, he is an artist, and is therefore poor.
There is a 2-foot diameter wooden ball, a 3-foot diameter bronze ball, and a 4-foot diameter concrete ball. How much paint (in terms of square inches covered) would be needed to cover them and have none left over?
This is really deceptively easy. Just plug the values into the formula:
BALL 1 (2 FOOT DIAMETER):
This one is 2 feet across, so we convert to inches (24) and half it (12) to get the radius in inches.
Then we plug in the formula values:
4 (pi) (r)2 (Where r is the radius)
4 (pi) 122
4 (pi) 144
12.566 (144)
1,809.56 square inches
Simple, no? Let's get the same thing going with the second one.
BALL 2 (3 FOOT DIAMETER):
This one is 3 feet, so we cut that in half and convert to inches again (18). And then plug it in:
4 (pi) (r)2
4 (pi) 182
4 (pi) 324
12.566 (324)
4,071.50 square inches
Now, I would walk you through the third one, but I think you get the idea. I will do the quick sum up.
BALL 3:
12.566 (24)2
12.566 (576)
7,238 square inches
So, add those up and you get the amount of square inches you need to cover:
1,809.56 + 4,071.50 + 7,238 = 13,119.06 square inches (or 91 square feet).
Now go buy that paint!
Wrapping Up
Now your artist friend is happy because his installation piece is done, and you're happy because you can get the surface area of a sphere! Everybody wins!
That's about it. The process is the same for any sphere, the only thing that might complicate it is having odd radius lengths (2.333333... inches or something like that), but that is what a trusty calculator is for. And if you can't use a calculator, you can always find an Instructable to show you what you need to do to get it figured out!
I hope you enjoyed this Instructable and it really made you understand the ins and outs of this idea. As always, comments are welcome! Now get out there and start calculating!
That's about it. The process is the same for any sphere, the only thing that might complicate it is having odd radius lengths (2.333333... inches or something like that), but that is what a trusty calculator is for. And if you can't use a calculator, you can always find an Instructable to show you what you need to do to get it figured out!
I hope you enjoyed this Instructable and it really made you understand the ins and outs of this idea. As always, comments are welcome! Now get out there and start calculating!