Volume of a wheel

Math has been an integral part of almost every culture from the time civilization began. From the ancient Indians, to the ancient Greeks, every civilization gave importance to math. As vast and diverse of a topic math is today, to them it was very important for them to solve practical problems and religious activities. Ancient mathematicians spent much of their time thinking about and solving problems related to geometry and arithmetic. Hence, many of the most famous concepts and innovations in math have their origin in geometry. This article attempts to help us understand how mathematicians have thought about solving problems and how that has led to the discovery of some of the most fundamental concepts in mathematics.

As a practical example let us imagine what ancient mathematicians reasoned when their local wheelwright asked them a question: How much wood do I need to build this wheel?

Let us follow a process they may have followed. We can start by simplifying this question. The amount of wood refers to the volume of a wheel. We can simplify a wheel into 2 main parts: a circle(frame) and its radius(spokes). For now, let’s ignore the cylinder at the center to which the spokes are attached. The radius is easy to calculate, we can just measure the length of the spoke. However, calculating the length of frame is not as straightforward. It is impractical to just measure the circumference by taking a measuring device and calculating its length. How do we make this process easier? If we think about it, a curve is just a line which has many bends. So, if we straighten the border out, it should be possible to convert the curve to a line? if we take a rope and turn it into a circle of the size of the wheel, then we may measure the length of rope and suddenly we have calculated the circumference. Just like that, we have derived a simple process to find the circumference of any circle using the same method.

We still need to calculate the volume of a wheel. To simplify this process, assume that spokes are cuboidal and simply multiply the thickness and height with its radius. Similarly calculate the volume of the frame by assuming that it too is cuboidal, and add the values and we have gotten a pretty good estimate for the volume of wood the wheelwright needs to procure to build his wheel. While we have answered the question, as a curious person we wonder, can we find a way of relating the length of spokes to the length of the frame to avoid tediously measuring the length the frame every time he gives me new dimensions of the wheel?

The most obvious thing to do would be to do compare the circumference and radius of a circle with radius 1 to the circumferences and radius of circles with differing radii and derive a relation. While doing this, we find that the circumference of a circle with radius 1 is equal to some fixed constant and that the radius of the other circles just scales this constant to get the circumference. Today we know that fixed constant to be 2π. Hence, we have proved that the circumference of a circle is 2πr. But notice, the method we used for calculating the volume is inaccurate. We have assumed the frame and spokes to be cuboidal. Looking at the image again, it looks like the spokes are more similar to cylinders than cuboids. Therefore, to make the model more accurate, we need to calculate the volume of a cylinder.

Let’s use the general method we have developed to get to this stage. Again, let’s start simplify the problem. We can think of a cylinder as many circles stacked upon each other. Therefore, the volume of a cylinder should be related to the area of a circle and its height. Now, we need to find the area of a circle. As shown in the diagram above, if we split the circle into an even number of slices with their edges equal to the radius of the circle and arrange the slices in the pattern depicted above, we get a rectangle. The sum of lengths of the curved part of each slice is the circumference of the circle, which we have derived to be 2πr. Since half of the slices are arranged on one side of the rectangle, the length of one side is half of the total or just πr. The area of the rectangle we get is πr^2 and hence, the area of a circle is also πr^2. By multiplying the area of a circle with its height, we get the volume of a cylinder as πr^2h.

We have now developed the tools to accurately calculate the volume of every part of the wheel and we can even account for the cylinder in the middle. Initially, we had split the wheel into 3 parts: the frame, the spokes and the cylinder which connected the spokes. All of these components are cylinders. To calculate the volume of the frame, we just subtract the volume of the inner rim from the outer rim. To calculate the volume of the spokes, we just calculate measure an individual spoke’s, radius and height and calculate its volume. Then we multiply the volume with the number of spokes. Finally for the volume of the cylinder in the center, we repeat the process we used to calculate the volume of an individual spoke. Note: Assume The height of both the outer and inner rims and the connector are the same. We add all of the above to get the following equation.

R1: Radius of outer rim; R2: Radius of inner rim; R3: Radius of Spoke; R4: Radius of connector; H: Height of wheel; h: Height of spoke; n: number of spokes

Just like that, we have managed to derive the formula for the circumference of a circle, area of a circle, volume of a cuboid and the volume of a wheel. In fact, Archimedes calculated π by considering a similar problem. Instead of this method, Archimedes used the length the wheel traveled before a particular spoke reached back to its starting point on the wheel to calculate the circumference. Meanwhile, ancients Indians calculated it while considering areas and circumference of altars.

If you are interested in calculus, another interesting thing to note is that if we integrate a function representing the circumference of a circle(y=2πr), we get the area of the circle(y=πr^2). Normally, to get the area of a circle, we integrate the equation of a circle(x^2+y^2=r^2). This fact begs the question, why does the integral of the circle’s circumference result in its area? To understand this, let’s we split up the circle into many rings with a very small thickness of ‘dr’ and we sum the areas of all the rings, we should get the area of a circle. This process is similar to just integrating the circumference. Therefore, the integral of the circumference is the area of the circle.

The point of this thought experiment was to show how rooted in reality many of the problems are that led to the discovery of fundamental mathematical concepts. Sometimes, while solving math problems we just blindly apply formulae and shuffle symbols to get the answer. While this is the fastest and most efficient way to solve most problems, if we never stop to check that what we are doing makes sense and that there is a sound, logical system behind it, our understanding of math is limited and we may miss the beauty in math. In fact, concepts fundamental to calculus such as integration, were first developed by considering similar such geometric problems.

Related posts:

We all know that MBBS is a noble profession. Because medical field is directly related to someone live So to get in this noble profession you need to clear your entrance test in pakistan known as…

Spring is an open-source framework based on the Java language, and here Controller is a class to handle requests coming from the client. Today we will learn about Spring Controllers and how to test…

if statement is the most basic of all the control statements. It tells your program to execute a certain section of code only if a particular test evaluates to true. An if statement consists of a…