Introduction: Volume of a Sphere
First Prize in the
Burning Questions Round 6.5
A sphere is nothing but a three dimensional circle. If you can imagine a smooth ball, suspended in mid-air, perfectly round... then you get a sphere.
All of you have seen spheres...They're everywhere!
If you've seen the solar system, you've seen spheres..If you've played ping pong, or golf or soccer...even basketball or cricket and hockey...you've seen spheres. If you are a housewife, you'll see spheres in the fruit basket... And if you are in kindergarten...they're in your abacus set!
Spheres are very common shapes, one encounters in everyday life.
This instructable will show you "how" and "why" to calculate the volume of a sphere.
I have explained every concept and every formula in great detail. Also , I have filled up this instructable with a lot of practical examples which will help you understand me
Step 1: Your Toolbox
If you're planning to do some real-time calculations, here's what you will definately need:
1) A good ruler.(Or if you have a good vernier calliper in the local store, that would be excellent!).
2) A long piece of thread( Extremely useful in finding the circumference!)
2) A good and reliable calculator.
3) A pencil or marker of some sort.
4) Paper on which you can do your calculations.
Step 2: Your Virtual Toolbox: the Circle!
When you're out calculating the volume of a sphere, it's important to have the right tools with you. In the last step, I explained what you need to put in your toolbox to get going. In this step I'll tell you some of the concepts you need to carry around in your "virtual" toolbox if you are dealing with spheres.
Maybe you already know about the circle...but I think that a quick revision of some of the important concepts would be a great idea!
1) The circleis the two dimensional form of a sphere. In simpler terms it is round and flat.
For example, there is a ROUND garden in the neighborhood. Then I can say that the garden is an example of a CIRCLE or in other words, the garden is CIRCULAR.
2) The radius of the circleis the distance from the centre of the circle to a point ON the circle.
For example, one boy runs from the centre of the garden to the edge of it in a straight line. The distance covered by the boy is equal to the radius of the garden!
3) The diameter is exactly double the radius.( it is the longest distance between any two points on the circle). In other words, if we draw a STRAIGHT line from any point on the circle to another point on the circle,and if the line passes through the centre of the circle, then the length of the line is the diameter of the circle.
4) The Circumference of a circle is the total length of the edge of the circle. That means, if I break the circle at any point, and spread it out on a piece of paper, as a line then the length of this line is known as the circumference.
Now let us solve an easy problem on this concept!
Question: I take a walk, every morning around a circular garden. I walk exactly once around the garden every day. If anybody asks me, I tell them that I walk one mile per day. What is the circumference of the garden?
Answer: If you have read the explanation of the circumference carefully, you will realize that the circumference of the garden is exactly one mile!
5) The Area Of the Circle is the total space occupied by a circle.
For example, If I have to mow a circular garden completely and I manage to cover the whole garden...then I can say that I have covered the entire AREA of the garden!!!
Coming next to you, is the MOST BEAUTIFUL NUMBER ON THE PLANET!!!
Step 3: Your Virtual Toolbox: Some Vital Tools!!!
Now, a very interesting value links the concepts of the circumference and the diameter together. It's very interesting to note that if I take The Circumference of the circle, and divide it by The Diameter, then it always will give me the same value no matter what dimensions the circle has!
This special value is known as PI(pronounced as "pie"), and it has the value 3.141592654....For simplifying the calculations, we assume the value of PI to be only 3.14.
This value PI is the most important tool we have, SO DON'T LOSE IT! IT's EXPENSIVE!
Now, combining everything that we learned in the last step and using the value of PI as a tool let's discuss some vital formulas!
(Please replace the * sign by multiplication signs during application)
1)Circumference of the circle = 2 * PI * radius.
2)Area of the circle = PI * radius* radius.
Using these concepts, let's calculate the radius, diameter and area of my garden...REMEMBER, as discussed in the last step, the circumference of my garden is "one" mile.
I leave the calculations to you ...but in order to assist you , I have made a table which will be of use to you in you calculations. After you are done with the mathematics , you should end up with the following figures:
Area= 0.0796 square. miles( REMEMBER AREA IS ALWAYS IN SQUARE TERMS)
Our basic toolbox is ready.
Now, we will construct special tools for the object, we are most concerned about................ ..........................................................THE SPHERE!!!
Step 4: Your Workshop: Building the SPHERE!
Now that we're equipped with all the tools, lets move on to THE WORKSHOP.
The workshop is where we analyze the object in concern, in this case , the sphere.It's a place where we put our tools to use!!!
I have mentioned before, that the sphere is one of the most common shapes that we find in everyday life.
Spheres have a characteristic compactness, which make them even nature's favorite shape.
1)Have you ever noticed a raindrop?
Have you observed that it is perfectly round in shape, i.e spherical in nature?
2)Have you seen an observatory? Or a planetarium? Have you noticed that the structure is exactly half a sphere?
3)Or maybe....the ball-bearing of a ball-point pen?
4)Or if you are more inclined towards biology...have you noticed our eyeballs??? Perfectly spherical?( In fact, the spherical nature makes it possible for us to see over such a wide range...Imagine a cube shaped eye trying to move around in the socket !!!)
5) Have you ever heard of the "Buckyball" or The "Fullerene"? Many a times , tiny carbon atoms bind together to form a perfect sphere. This sphere is a nanoparticle or a "nanosphere"!
How many such spheres can you see in day to day life....can you find the reason behind their spherical shape?
Let us now understand the important concepts linked with the sphere... You will be surprised to note that all the concepts that we studied in the Circle ( step 3) apply to the spheres (with just a bit of tweaking!) !
1) The sphereis the three dimensional form of a circle.
In my neighborhood, there is an old fortune-teller. She owns a big "orb" in which she sees the future! If all of you have ever seen an orb,you will have noted that the "orb" is spherical in shape.
2) The radius of the sphereis the distance from the centre of the sphere to a point ON the surface of the sphere.
3) The diameter is exactly double the radius.( it is the longest distance between any two points on the sphere). In other words, if we draw a STRAIGHT line from any point on the sphere to another point on the sphere,and if the line passes through the centre of the sphere, then the length of the line is the diameter of the sphere.
4) The Surface Area Of the Sphere is the total area occupied by the surface of the sphere.
In other words, if I cover my sphere with a cloth which covers it perfectly, then i can say that the area of the cloth is equal to the surface area of the sphere!
With this, we cover all the basic concepts related to spheres....
In the next step, we learn how to use our tools for the final kill, i.e finding the volume of any sphere!
Step 5: The Laboratory :Volume of the Sphere.
WE researched in "step 2" of this instructable, that the amount of space occupied by a circle is known as the "AREA" of the circle. I also gave an example of mowing a circular garden.
But when I define a sphere, which is basically a 3 dimensional object, what term do I use for the space occupied by the 3-D structure?
And therefore we use a term called Volume ,to define such a mathematical quantity.
Thus, volume of the sphere is nothing but the space occupied by a sphere!
In other words, what area is to 2-D, volume is to 3-D.
Thus in order to calculate volume:
Step 1) We take the "'area" of the 2-D equivalent of the shape.
Step 2)We add a third dimension to this area!
Step 3) Voila! We have the volume!!!
The method by which we follow the above procedure is called The Method Of Integration. It gives the formulas of volumes of ALL different 3-D shapes!
So by the method of integration, we get a formula for the volume of the sphere.
Volume of the Sphere= (4/3) * PI * radius * radius * radius
I have disintegrated the above formula and I've turned the formula into a set of instructions, which give us the volume in a much simpler manner....
Now, how do we go about actually calculating the volume of the sphere???
Step 1 Take a fine thread and hold it's end on the sphere firmly! Then take the thread once round the whole sphere till you reach the first end of the thread.
Step 2 Mark the point on the thread which touches the first end.
Step 3 Now measure the length of the thread from one end till the marked end. This is the circumference of the sphere!
Step 4 Calculate the diameter of the sphere. ( or if you have a vernier calliper, you can find the diameter directly!)
Step 5 Divide the diameter by two. We get the radius of the sphere
Step 6 Take the cube of the radius.( in simple words, "radius" multiplied by "radius" multiplied by "radius").
Step 7 Multiply the above answer with PI. ( i.e 3.14).
Step 8 Multiply the above answer by 4 and then divide the answer obtained by 3.
The answer you will obtain in the above step is the volume of the sphere.
1) REMEMBER that the "unit of the volume of a shape" is the CUBE of the "unit of the radius". So, if the input radius is in (metres), then the volume will be in (metres cubed) .
2) REMEMBER that if, for example, the unit of the radius is metres, then the unit of the volume has to be (metres cubed) it cannot be (centimetres cubed) or (inches cubed) unless of course if you have yourself changed the unit using conversions.
3) IN CASE YOU FORGET THE EIGHT STEPS I HAVE MENTIONED ABOVE,
Have a look at the table I have provided below. If possible , take a print-out of it.... One look at it will summarize all the 6 steps!!!
But why do we need this particular term called volume? Does it have any practical applications?
In the next step,we will solve some real-time examples of the concept of volume to get a better idea!
Step 6: Applications of the Volume Formula.
We're out there in the real world now. With a handy little toolbox and some useful tips, lets try and tackle some problems...
Imagine you are in the 16th century. Italy. Sitting at the desk of Galileo Galilei. Galileo has given you the task of finding out the volumes of certain planets. He has provided you with the radii of those planets. They are as follows:
Venus------ 6,051,000 metres.
Mars ----- 11145013.12 feet.
Jupiter----- 78184601.92 yards.
Galileo wants all answers in terms of cubed metres . What answers will you give Galileo?
SPOILER: Watch out for Mars and Jupiter!
Well, since we know the radii ...the problem becomes very simple!
The Answers are:
Mercury---------6.085 multiplied by 10 raised to the power 19.
Venus----------9.28 multiplied by 10 raised to the power 20.
Mars------------1.642 multiplied by 10 raised to the power 20.
Jupiter--------- 1.5306 multiplied by 10 raised to the power 24.
(All values are in cubed metres.)
I have to build a semi- spherical planetarium inside a huge room of volume 63,000 cu.metres. The diameter of the planetarium should be 50 metres. Will my planetarium fit inside the room?
ANSWER: Remember that the planetarium is SEMI-spherical. So it's volume is half the volume that we calculate for a sphere. Half the sphere will quite easily fit inside the room.
Volume of the planetarium=32724.92 cu.metres.
PROBLEM 3) Imagine that one day Mr. Gates, the millionaire goes mad and decides to fund a project in which the moon is to be opened up and completely filled with marbles having circumference 2metres. Now the radius of the moon is 1,738,000 metres. Approximate how many marbles will fit into the moon till the moon fills up completely!
The no. of marbles =(volume of the moon) divided by (volume of each marble).
And radius = (circumference)/ 2*PI
Volume of each marble= 0.1351 cu.metres
Volume of the moon =2.199 multiplied by 10 raised to the power 19.
The no. of marbles =1.6277 multiplied by 10 raised to the power 20!!!
NOTE: Given below, are the tables which summarize all that we have already examined in the steps prior to this one. They may prove to be useful in solving the above problems.
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