Could someone please check if I am correct? Thanks!

[Muppers3262]

Please check to see if I answered this question correctly because I'm not sure I did it right. I think the answer is the circle, but I'm not really sure. Thanks.

The circumfrence of a circle and the perimeter of a square are the same. Which has the bigger area, the circle or the square?

Since the circumfrence of a circle is 2*pi*r and its area is pi*r^2. While the perimeter of a square is 4x and its area is x^2.

 

[Euler]

Ignore

 

[ting]

For a given perimeter the circle has the greatest area of any regular polygon. As the number of sides of a regular n-sided polygon approaches infinity, its area approaches the area of a circle with the same perimeter.

If the circumference of a circle and the perimeter of a square are the same then the circle has the larger area.

The area of a square with a perimeter of 1 is:

= 0.25^2

= 0.0625

The area of a circle with a circumference of 1 is:

= 1/(4pi)

= 0.0796

~~~~~~~~~~~~~~~~
The area of the circle is (pi)r^2

Find the radius from the cicumference

2(pi)r = 1

r = 1/2*pi)

Substitute into area formula.


A = pi * [1/(2*pi)]^2

A = pi * 1/(4(pi)^2

A = 1/(4*pi )

= 0.0796

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Edit: looking at your other posts you may want some more detail.

The area of a regular polygon is given by:

A = (1/2)*p*r

Where, p, is the perimeter and r, is the radius of the incsribed circle.

From this you can get the following formula for area of a polygon:

(p^2)/(4ntan(pi/n)

Where p is the perimeter and n is the number of sides.

And

For the circle substituting, p = c = 2*pi*r, into

A = (1/2)*p*r

gives:

A = pi*r^2

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

 

[Muppers3262]

Thanks Ting, for explaining why the circle had the larger area.