By Jonathan Lupercio

In this article we show you how to derive the famous equation for the area of a circle. Most students simply memorize the formula and plug in numbers in order to calculuate the area of a circle. It's unfortunate that this derivation is largely left out of the standard math curriculum, as this is where the joy is found! We will show you how to derive the forumla for the area of a circle using the same methods as the ancients Greeks.

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The Area of a Circle

    Background

    Measuring the area of objects was an important task in the past. For shapes with straight edges, it was simple to break them down into squares, rectangles, or triangles. However, one shape that posed a problem for mathematicians was the Circle. With no straight edges, all mathematicians could do was come up with an approximation. But before calculating the area of a circle, they would first need to determine the circumference of a circle.

    Circumference

    The circumference is a special term given to the perimeter of a circle. For most shapes, it is simply the sum of the outer lengths of a shape. The circle posed a problem for its singular, curved side. The best way to figure out the circumference would be to take a rope or string, wrap it around a circle, and then measure the length with a straight ruler. As mathematicians, this is not an exact formula!  We would prefer to calculate the area of a circle with a diameter, which is a more fundamental and measurable quantity.

    Diameter

    The diameter is the longest distance from one edge to its opposite point. Every circle has one, and different circles can be differentiated based on their diameter.  Therefore calculating the area of a circle with a diameter is a good candidate. Some of you may ask, “Why not the radius?”, and that is an excellent question! The reason is it would be difficult to pinpoint the exact center of a circle, especially by eye.  There is much less room for error in measuring the diameter. It was also thought of as the “width” of a circle. Fast forward some centuries and all we need to do is divide the diameter by 2 to calculate the radius.

    $D=2r$

    The Ratio of the Circumference to the Diameter

    The question now becomes, how is the diameter related to the circumference C? People would measure the diameter of a circle, and then the circumference to see if they are related. After many careful measurements of circles of various sizes, they would determine this relationship: the ratio of the circumference to the diameter was a constant!

    $\frac{C}{D}=3.14159$

    This means that the circumference was a little over 3 times larger than the diameter. This new constant would be famously called Pi (π). By writing it more cleanly we get

    $C=πD=2πr$

    We would obtain the dependence on the radius by substituting our earlier equation relating the diameter to the radius, which is the form mostly used today. For centuries, mathematicians attempted to precisely calculate the exact value of π, only to find that it is a non-repeating, never-ending number! We would later find that it is an irrational number, but that topic will be for another time.

    Area of a Circle

    Now we can begin to discover the formula for the area of a circle. Firstly, let us start with the area of a triangle. To find the area of a triangle we need two pieces of information, the length of its base and the height of the triangle measured perpendicularly from its base.

    The formula is given as

    $A_{∆}=\frac{1}{2}bh$

    The approach we will take is first approximating the area using triangles, then using smaller and smaller triangles to better approximate the area of a circle, giving us a precise answer at the end. We can begin by using n number of identical isosceles triangles. In the picture below we use n=12 triangles.

    Screenshot-2023-11-30-at-6.22.55 PM

    We can now write the sum of the areas of the triangles as

    $A_{o}=n*(\frac{1}{2}bh)$

    This equation simply multiplies the number of triangles n by the area of a triangle, giving us the area of all the triangles we used. Plugging in n=12 we get

    $A_{o}=(12)*(\frac{1}{2}bh)$

    This is the best we can do for now, however there are some observations we can make. The height h of the triangles runs from the center of the circle perpendicular to each base of the triangle. The bases b of each triangle are the sides running around the outer edge of the circle. What happens if we start making more and more smaller triangles? Let’s see.

    area of a circle

      Now we have n as a really large number. What do we notice? Firstly, we see that the height of each triangle now is approaching the same length as the radius! Also, what can we see about the bases? That’s right, the sum of the bases of all the triangles is becoming the Circumference! So, if we take the limit as n triangles becomes infinitely large, that is make n as large as possible to the point that the observations we made become true, we can finally write

    $A_{o}=n*(\frac{1}{2}bh)$

    $A_{o}=(nb)*(\frac{1}{2}h)$

    In this line we substitute the observations we made, replacing the base of each triangle times the number of triangles nb which becomes the circumference 2πr, and the height of each triangle h with the radius of the circle r

    $A_{o}=(2πr)(\frac{1}{2}r)$

    From here we can do some simplification. The 1/2 cancels with the 2 from the circumference, and the radius is squared. This leaves us with the equation

    $A_{o}= πr^2$

    This is the famous Area of a Circle equation!

    Closing Thoughts

    Now you know where the formula for the area of a circle comes from!  Isn't this much better than simply memorizing the formula?  Almost every student learns how to calculate the area of a circle by simply plugging in numbers.  Here at Alexander Tutoring, we make sure our students can derive every formula they use.  This is where you will find the deeper meaning behind the equations, which greatly increases engagement and joy.

    Though the ancient Greeks had many remarkable advances in Geometry, the circle is one shape that gave them a particularly difficult time. In the third century B.C., Archimedes inscribed and circumscribed regular polygons on the inside and outside of circles to approximate the circumference. He managed to calculate π to three digits, 3.14. The mathematical tools and techniques available to people 2000 years ago severely limited how accurately they could calculate π. Fast forwarding to the 16th century, the development of infinite series allowed mathematicians to approximate π to 12 digits, and then further in the following centuries. In the mid-1700s Johann Lambert proved that π is indeed an irrational number, never repeating and never ending. Today with the invention of computers, π can be calculated to as many decimal places as you like! All of this ultimately leads to a more and more precise calculation for both the circumference and area of a circle.

    Author

    • Jonathan Lupercio

      Having grown up in the Hamptons of Long Island, a physicist is not the first profession that comes to mind. However, Jonathan had developed a passion for science and mathematics at an early age, seeing them as universal languages. He excelled in physics and knew early on that he would pursue it at university. He attended Stony Brook University for its rich and rigorous physics program, as well as minoring in Mathematics. He joined Professor Marivi Fernandez-Serra on her research team after acing her courses in Scientific Programming and Thermodynamics by the end of his sophomore year. There he learned more advanced programming methods centered around molecular simulations. The team's research focused on simulating icing models of water. Afterward, Jonathan dedicated some of his time to volunteering and tutoring other students in the Physics Help Room, which he himself had used in previous years. Having had many great professors and instructors, he took the best lessons and helped others. He discovered an ability to communicate complicated subjects in an easier and more understandable approach. His core teaching principle is that of intuition, focusing heavily on ensuring physics and mathematics come naturally in the thinking of his students.

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