How do you prove that the area of a circle is equal to pi*r^2?

Question

It looks right visually but I wonder how to prove it. (Don't worry, I'm no longer in school so it's not a homework problem. I just really miss math.)

Answer 1

Divide your circle into an even number sectors of equal sizes (like pie pieces), and roll them out. Take half the sectors and turn them over (rotate 180 degrees) and place on top of the first half of sectors and adjoin the straight edges. This resembles a rectangle. The circumference of a circle is pi times the diameter, so this mean that one side of this 'rectangle' is pi times half the diameter or pi times the radius. The height of this 'rectangle' is the radius of the circle. Now, if you create an increasingly large number of sectors, the figure becomes more and more rectangular.. and if you could produce an infinite number of sectors, then it would actually become a rectangle.. and the area would be exactly pi times the radius squared.

The link gives you a visual of the sectors...

Answer 2

By the way, circles don't have area. Circles are defined as the set of points which are all an equal distance away from a center point. You more correctly have to talk about the area inside a circle, which is called a disk.

There is a calculus proof of that. Calculus is good for figuring out slopes, rates of change, and areas. It asks you to visualize infinitely small segments and shapes. The math theory is proven that, "in the limit" if you go infinitely small, the area totals to the right answer.

To prove the area of a circle, divide your circle into an infinite number of tiny triangles and unroll the circle so the outside of the circle is now perfectly flat, "on the floor".

Each triangle is as high as the radius of the original circle, r. The area of a triangle is equal to 1/2 times the base, times the height. It doesn't even matter if the triangle stands straight up. It can be skewed, but as long as you measure where the height ends up (measured at right angles to the floor), the area is still the same.

Now, those bases on the infinitely small triangle are also infinitely small, hard to work with, but if all the triangles are the same height, you just add up the bases to get the total area of all triangles. The total length of the triangle bases are equal to the circumference of the original circle. Finite slices of a circle give "wedges" that actually have a rounded base when you unroll the circle, but "in the limit" the base of these triangles is perfectly flat.

So the area of the circle has to be: 1/2 * total length of base * height

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

Has to be!


Wanna go 3-D? You can prove the same sort of thing for the volume of a sphere. The inside has to be divided into infinitely small pyramids, and pyramids only have 1/3 of the volume of a cube of the same height.

Surface area of a sphere is 4 * pi * r^2

Volume of sphere = 1/3 * total area of base * height
= 1/3 * (4*pi*r^2) (r)
= 4/3 pi r^3

Answer 3

Start by drawing a regular polygon. An octagon would be a good starting point. How would you find the area of this? One way is to slice it into 8 congruent triangles, then find the area of one of those triangles. The base of the triangle is one of the octagon sides, and the height of the triangle is the distance from the side to the center of the octagon (known as the "apothem").

Let s = side of octagon, and let a = apothem of octagon.

The area of each triangle is (base)(height)/2 = (s)(a)/2.
Multiply by 8 to get the area of the whole octagon:

A = (8)(s)(a)/2. But (8)(s) is the perimeter of the octagon. Letting P=perimeter of the octagon, we have this general formula:

A = Pa/2. This formula actually applies to any regular polygon, since the number of sides will always be equal to the number of triangles. Multiplying the base of a triangle by the number of sides will always give us the perimeter of the polygon.

Now imagine drawing a regular polygon with 100 sides. This would look a lot more like a circle, wouldn't it? 1000 sides--even more so. 10,000 sides? No different than a circle as far as the eye is concerned. As you get more and more sides, the polygon approaches the appearance of a circle more and more. The perimeter of the polygon approaches the circumference of a circle, and the apothem of the polygon approaches the radius of a circle. Substituting the parts of a circle into the area formula gives

A = Cr/2, where C = circumference and r = radius.

Since C = 2(pi)(r), we have

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

I have questioned this as well, and this is the reasoning that satisfies me the most. Hope it helps! (P.S. My wife thinks I'm an absolute nerd now).

Answer 4

You are confused. The sum is wrong. It can be solved only if the word circle is replace by " sphere ". In that case Let R be radius of the sphere, and x be the side measure of the cube. then - Surface area of - The sphere = 4.Pi.R^2 and that of - Cube = 6 x.R^2 But if surface areas are equal to each other, then 4.Pi.R^2 = 6 x.R^2 => (R/x)^2 = 3/(2.Pi) => (R/x) = Sqrt of 3/(2.Pi) => (R/x)^3 = [3/(2.Pi)]^3 Now Volume of - Sphere (V1) = 4/3 . Pi. R^3 and that of Cube (V2) = x^3 Therefore, (V1/V2) = (4.Pi / 3) (R/x)^3 = (4.Pi / 3) (sqrt of (3/2.Pi)^3 ie (V1/V2) = (4Pi/3) . [ (3.sqrt 3) / (2.Pi . sqrt 2.Pi ) ] => 1 / sqrt of ( Pi / 6 ) ................................ Answer

Answer 5

Draw a circle centre O and radius r.
Within this circle draw another circle of radius x
and another circle of radius x + ðx where ðx is a small increment.
Area of strip of thickness ðx = 2πx.ðx
Now have to sum all such strips within the circle to give A, the area of the circle:-
A = ∫ 2π x dx between limits 0 to r
A = 2π.x²/2 between limits 0 to r
A = π.r² as required.

Answer 6

RE: the replies that state the solution requires calculus, I thought the area of a disk formula dated back to ancient Greece, while calculus I understood only dates back to Sir Isaac. How do you suppose the Greeks worked it out, or does calculus predate Isaac Newton?

Answer 7

if you'd like a simple geometric approximation of the area proof, you can check out this site

http://www.mathreference.com/geo,circle.html

if you'd like to use a bit of calculus to actually derive it, check this out (it's a pdf)

www.artofproblemsolving.com/LaTeX/Examples/AreaOfACircle.pdf

sorry that last link doesnt look as if it came out right, ill put some space into it so you can cut and paste in your browser

www.artofproblemsolving.com/
LaTeX/Examples/AreaOfACircle.
pdf

Answer 8

You know pir^2. Did you realize the derivative of that is 2pir.

To answer your question:
http://curvebank.calstatela.edu/circle/circle.htm theres a good link, even has pictures!

Answer 9

I'm sure there is a proof for it somewhere. It's been so long.. this question remind me of when I had to write up a 4 page proof of why zero is not a number.. You'll have to look it up for the proof. Can't type it here anyhow..

Answer 10

Its great you have enthusiasm and curiosity for math.

This link contains the proof for area of a circle.

http://www.artofproblemsolving.com/LaTeX/Examples/AreaOfACircle.pdf

It requires calculus and trigonometry, but is straightforward if you go step-by-step.