(No Greek letters were harmed in the writing of this blogpost.)
πr2 is the area of a circle, but for a lot of people, this is another one of these ‘faith’ formulae. So here, I offer a proof of this fact:
It is known that x2 + y2 = r2 is the equation of a circle. (For an explanation of this, see this site.)
The circumference, by definition, is 2π times the radius. So, circumference = 2πr.
(I can justify this by saying that π = circumference/diameter = c/2r. So c = 2r*π = 2πr.)
Now, imagine a small part of this circle, like this (but with a far smaller angle):
The white line to the left of the circumference is y, and the angle is θ.
Circumference of wedge = 2 πr * (θ/360)
It follows that if is very small, c = y. That means that the wedge is nothing but a simple triangle.
y = 2 πr * (θ/360)
So, the area of the triangle is ½ yx.
Area2 = x2y2/4
= (x2/4) * (2πr * (θ/360))2
= x2 π2r2 (θ/360)2
Therefore, area = πrx (θ/360)
Since this triangle is right-angled (although, thanks to my poor MS Paint drawing skills, it doesn’t look like it) at the vertex common to x and y, you can say that cos = x/r, and so, it follows that x = r cos θ. So, if you substitute r cos θ in the formula, you get:
πr2 (θ/360) cos θ
This is the formula for the area of a circle wedge, of angle θ. So, if you work it out for θ = 360 degrees, you get:
πr2 (360/360) cos θ = πr2 cos 360.
cos 360 = 1, so the area of a circle of radius r is πr2.
With this formula, it’s possible to express the area of a circle in other ways, too! I’ll show you two of them.
c = 2πr = πd.
A = πr2 = π(2r)2/4 = πd2/4 = cd/4.
This formula completely avoids π!
The second one is one which I derived myself, and I could not find any mention of it until recently, when I Googled it and found that I was not the first one (as it had happened before).
I derived this at the age of 10, when there were lots of maths questions of the form “Here is the circumference. Find the area.” or vice-versa. The ‘proper method’ (one thing which I have never been very good at sticking to) was to divide by , find the radius, then square it and multiply by to get the area. To me, it seemed inefficient, so I worked out a formula that used the circumference to directly find the area:
c = 2πr
c2 = 4π2r2
c2/4π = πr2
This is a formula that I haven’t found in any books or similar resources – it’s one of those secrets of maths, really.