Finding the Area of a Circle

Students often wonder where the formula for the area of a circle comes from; and knowing something about that can help make it more memorable, as I discussed previously about other basic area formulas.

Who first used the formula?

Let’s start by taking a historical look:

History of Circle Area Formula

Do we know who figured out that pi r squared is the area of a circle?

I can find out about the history of Pi and the circumference of a circle, but not its area.  I looked through your FAQs and on Google but to no avail.  Perhaps it is just not known?

This was a tricky question to answer, because the very idea of formulas came long after the first people to find areas. I pointed to an early analogue of the formula:

It's hard to answer that question, because the area of a circle was known long before pi was actually used.  Proposition 2 of book XII of Euclid's Elements, which was undoubtedly known before Euclid himself, is equivalent to the formula A = pi r^2:

  Euclid's Elements Book XII, Proposition 2  
http://aleph0.clarku.edu/~djoyce/java/elements/bookXII/propXII2.html 

  Circles are to one another as the squares on their diameters.

That is, the area of a circle is proportional to (2r)^2, which in turn is proportional to r^2.  All that is lacking here is a name for the constant of proportionality, which has been called pi since 1706.

I could also have mentioned an ancient Egyptian method that comes a little closer to being a formula, which I find described here and here, taken from the Rhind papyrus. (We had answered a question about it here.) As a formula, we would express it as \(A = \left(d – \frac{d}{9}\right)^2\); in words, as translated in the first reference, it looks like this:

Example of finding the area of a round field with a diameter of 9 khet. What is its area?

Take away 1/9 of its diameter, namely 1. The remainder is 8. Multiply 8 times, making 64.

Therefore the area is 64 setjat.

(Early math was described by example like this, rather than being stated as a general formula.)

But I moved on to the recognition that the number used is the same pi that is used to find the circumference:

There are two parts to your question: who discovered that the area is SOMETHING times the square of the radius (for which the answer is whoever gave Euclid his proof, commonly considered to be Eudoxus); and who discovered that the constant of proportionality is pi.  The answer to the latter question is Archimedes.

The form in which Archimedes stated it was that the area of a circle is equal to that of a right triangle whose base is the circumference of the circle, and whose height is the radius of the circle.  That is,

  A = 1/2 (2 pi r) r = pi r^2

in modern terms.  So except for the lack of algebraic notation and a name for pi, he got the entire formula.  You may be aware that he also worked out the value of pi.

This is really somewhat remarkable; pi kills two birds with one stone! I gave a reference to the proof of this fact; we have also discussed it on our site, here:

Archimedes and the Area of a Circle

Deriving the formula as a rectangle

We have discussed various derivations of the area formula many times; I will show a few different ways here. First, from 2000:

Deriving the Area Formula for a Circle

Why is the area of a circle the square of the radius times pi?

Doctor Floor answered this time, including a picture:

Let's consider a circle with radius r.

If we divide the circle into an even number of sectors, we can rearrange these sectors as in the following figure:



The result is a sort of wrongly formed rectangle, but we know that the shorter "side" of this rectangle has length r, and that the longer "side" is half the perimeter, hence pi*r.

The more sectors we make, the more accurate our rectangle becomes. We can imagine that if we divided the circle into an infinite number of sectors, it would become a rectangle.

All derivations of this sort that we give fall short of being actual proofs, because we have to imagine what would happen for infinitely many pieces. Ultimately, this requires calculus to make it rigorous, though early versions (such as Archimedes’) used it in a disguised form. Our goal here is just to see that it makes sense. In the picture above, with more pieces, the sides would slope more and more steeply, approaching the vertical, and the top and bottom would become more and more flat, approaching horizontal lines. The area then will be the height (r), times the base (half the circumference):

Whatever the number of sectors we use, the "side" lengths will remain r and pi*r. Therefore our limit case with an infinite number of sectors still has sides r and pi*r, and the area of this limit rectangle is pi*r^2. Since the area of the circle does not change when we divide it into parts, the area of the circle must have been pi*r^2, too.

Doctor Dotty gave a longer version of this demonstration here:

Circle Formulas: Area and Perimeter

Deriving the formula using triangles

A similar demonstration can be done without making a rectangle:

Formula for the Area of a Circle

How do you get the area of a circle?

I haven't figured any of it out, but I want to know how to do it. Please help.

I started by showing the formula:

Hi, Kismet. I'm not sure whether you're asking for the formula for the area of a circle, or for an explanation of how it works. I'll give you both.

The formula is very simple:

    A = pi * r^2

which means the area is Pi (3.14159...) times the square of the radius. In a book it would look more like this:

        __  2
    A = || r

To use this formula, just measure the radius of the circle (which is half the diameter), square it (multiply it by itself), and then multiply the result by 3.14.

Then, to show that the formula didn’t come from thin air, I showed a way to think of it:

There's an interesting way to see why this is true, which may help you remember it. (Though the easiest way to remember the formula is the old joke: "Why do they say 'pie are square' when pies are round?")

Picture a circle as a slice of lemon with lots of sections (I'll only show 6 sections, but you should imagine as many as possible):

         *       *
      *   \     /   *
     *     \   /     *
    *       \ /       *
    *--------+--------*
    *       / \       *
     *     /   \     *
      *   /     \   *
         *       *

Now cut it along a radius and unroll it:

        /\      /\      /\      /\      /\      /\
       /  \    /  \    /  \    /  \    /  \    /  \
      /    \  /    \  /    \  /    \  /    \  /    \
     /      \/      \/      \/      \/      \/      \
    **************************************************

All those sections (technically called sectors of the circle) are close enough to triangles (if you make enough of them) that we can use the triangle formula to figure out their area; all together they are

    A = 1/2 b * h = 1/2 C * r

since the total base length is C, the circumference of the circle, and the height of all the triangles is r, the radius (if the triangles are thin enough). You should know that the circumference is pi times the diameter, or

    C = 2 * pi * r

(this is actually the definition of pi), so the area is just

    A = 1/2 (2 * pi * r) * r = pi * r^2

In other words, the area of a circle is just the area of a triangle whose base is the circumference of the circle, and whose height is the radius of the circle.

This version of the formula can be very memorable. I find that many students have learned the standard formula almost as an incantation — though many forget whether it is the formula for area or circumference. Relating it visually to triangle areas may help.

In my explanation, I wanted to avoid deep algebra (because the student was young), so I glossed over a detail I might have clarified. Properly, I should have shown why we can just use the total base length as if it were one triangle. That amounts to the distributive property: \(A = \sum \frac{1}{2} b_n h =  \frac{1}{2} \left(\sum b_n \right) h = \frac{1}{2} C r\).

I concluded,

What I've just done gets pretty close to algebra, which you haven't learned yet, but if you think about it (and maybe try actually measuring some real circles, or even make some lemonade) you should be able to see what I mean.
 
You probably didn't know that the area of a circle is the same as the area of a triangle!

Doctor Jerry gave essentially the same derivation here:

Why is Area of a Circle Equal to Pi * (Radius Squared)?

Deriving the formula using regular polygons

My favorite derivation comes by way of a formula that applies to any regular polygon, and also answers the question, How can you find the area of a circle without using pi?

Why Pi?

Dr. Math, I was just wondering... 

Why do we use pi when we calculate the circumference and area of a circle? I think one of my professors once told my class but I can't remember and am curious.

I first dealt with the circumference question, which is both simple and subtle:

Hi, Crystal.

There are two questions here, with very different answers.

First, for the circumference, it's because we DEFINE pi as C/D, so we can write C = pi D automatically. There's a trick hidden behind that definition, though: how do we know that C/D is the same for every circle? That takes a bit of proof, and leads to some interesting ideas; look in the Dr. Math FAQ on pi:

  Pi = 3.14159...
  http://mathforum.org/dr.math/faq/faq.pi.html   

or in the following answers in particular, for an explanation:

  Why is Pi a Constant?
  http://mathforum.org/library/drmath/view/57828.html   

  Einstein, Curved Space, and Pi
  http://mathforum.org/library/drmath/view/55198.html   

  Is Pi a Constant in Non-Euclidean Geometry?
  http://mathforum.org/library/drmath/view/55021.html   

Because the ratio of circumference to diameter is a constant (as long as we are working in a plane), we can give it a name (pi) and then use that definition to find circumference. Given the diameter, we use \(C = \pi D\); or, given the radius, we use the fact that \(D = 2r\) and substitute, so that \(C = 2 \pi r\).

Area, though, is a very different matter.

For the area, there's a nice way to see why the formula should be what it is.

Let's think about regular polygons first, and look at the relation between their areas and perimeters. Any n-sided polygon can be broken into n isosceles triangles like this:

        +-----+
       / \   / \
      /   \ /   \
     +-----+-----+      +
      \   / \   /      /|\
       \ /   \ /      / |a\
        +-----+      +--+--+
                        s

Each of these triangles has a base that is equal to a side s of the polygon, and a height a (called the apothem); the total area is

    A = n * sa/2

This can be rearranged as

    A = (ns)a/2

and since ns is just the perimeter P of the polygon, this means

    A = Pa/2

That can be a very useful formula in itself; it looks a lot like the formula for a triangle. But we can apply this to a circle:

Now make n very large, and a will be very close to the radius r of the circle the polygon is becoming. We can see (and could prove more carefully if we took the time) that for a circle,

    A = Cr/2

where C is the circumference (perimeter) and r is the radius.

But since we know

    C = 2 pi r

this becomes

    A = 2 pi r * r/2 = pi r^2

We're done! Because we could find the area of a polygon using its perimeter, we can find the area of a circle using its circumference, and that uses pi.

This approach lends can be turned into a real proof more easily than the others, which is why I like it more. There is no hand-waving about a curved figure becoming a rectangle or triangle. Yet the basic idea is identical to the methods I showed first.

This formula \(A = \frac {Cr}{2}\) is what I referred to earlier, an area formula that doesn’t use pi. It showed up in the triangle derivation above, and is the answer to the question posed here:

Finding a Circle's Area Without Pi

For a longer version of the same approach, using trigonometry, see

Areas of N-Sided Regular Polygon and Circle

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