can you proof a area of a square is equal to that of a circle?

Answer 1

[Edit: Those who say this can't be done because pi, hence sqrt(pi), is irrational, they're wrong. Because if they were right, it would be impossible to calculate the area of a circle, given the radius, for the same reason.

[If the geometry book says to calculate the area of a circle with radius 3, people will happily write 9 pi, or maybe 28.274...

[But if you can do that, you can just as well say s = 3 sqrt(pi) = 5.317... where s is the side of a square.]

We know that the area of a square is s^2, where s is the length of a side. And we know that the area of a circle is pi r^2, where r is the radius.

If A is the area of a circle, and A is the area of a square, then:

A(circle) = pi r^2 = A(square) = s^2

pi r^2 = s^2

r = s/sqrt(pi), or

s = r sqrt(pi)

Now do it in reverse. If the side of a square equals the radius of a circle times the square root of pi, then it's easy to prove that the area of the square equals the area of the circle.

Or if the radius of a circle equals the side of a square divided by the square root of pi, then the area of the circle equals the area of the square.

Answer 2

If you have a square of known area, you can easily work out the diameter of a circle that would have the same area - you just multiply the square area by 4, divide that by pi and take the square root of the answer. (Or just multiply the length of your square's side by 1.28379167.....).

However, if you have a circle and you want to construct a square of the same area using basic geometric principles, i.e. with a compass or dividers and a straight-edge you are out of luck. That's called "squaring the circle" and many ancient geometers drove themselves up the wall trying to do it - until analytical geometry and algebra produced the proof as to why it is impossible.

Another one is trisecting any angle from basic geometry, without angular measurement.

Answer 3

If S is a square, and "a" is the lenght of it's side, and C is a circle, being "r" it's radius, then:

As = a²
Ac = π r²

Where As and Ac stand respectively for the area of the square and the area of the circle. They are equal if, and only if:

a² = π r²
a = r √π

That is, the areas of a circle and a square equal only if the lenght of the square side equals √π times the radius of the circle.

Answer 4

A square of side length = a * sqrt(pi) is equal in area to a circle of radius = a

Answer 5

the idea behind this question is that (as someone else showed) the area of a square can only be the same as that of a circle if the side length is a multiple of sqrt of pi.
As pi is irrational i.e. a never ending decimal you can't draw a line with exactly that length, so you can't draw a square with exactly the same area.

Answer 6

In general, they will NOT be the same area (because you have said nothing about their sizes!)

If their dimensions were exactly the right ratio, they'd be the same size.

Circle area = pi*r^2 where r = radius
Square area = s^2 where s = length of one side
so s = sqrt(pi)*r for equal area figures.

Answer 7

well i will answer generally saying
area of circle = area of square
Pi x radius2 = (side of square)2
and solve for r on both sides??

Answer 8

I'm almost sure that it's unproofable, it's one of the oldest unsolved problem of mathematic.

Answer 9

I wish i stuck in at school. Aint got a clue what your talking about. Sorry,lol...

Answer 10

Algebraically or numerically?