# Construction Rectangle in Circle

• February 9th 2011, 10:56 AM
bram kierkels
Construction Rectangle in Circle
Given a circle and a square next to the circle . How can I construct a rectangle with the same surface as the square in the circle? Only using a compass and ruler.

Thanks
• February 10th 2011, 09:02 AM
Unknown008
Measure the length of one side of the square.

The length of the rectangle will be that length, x, multiplied by a certain constant k you don't know.

So, l = kx
The width will be w = x/k

So that the area of the square is x^2 and the area of the rectangle is (kx)(x/k) = x^2

In your circle of radius r that you measure, you will find that:

$l^2 + w^2 = (2r)^2$

Substitute l and w.

$(kx)^2 + \left(\dfrac{x}{k}\right)^2 = 4r^2$

Plug in the values of x and r to find k.

From there, you can get l and w, the dimensions of the rectangle to be drawn in your circle.
• February 13th 2011, 11:27 AM
bram kierkels
Thanks Unknown008. But I ment a ruler without numbers. So this should be a question in the topic "Euclidean Geometry". Can you help?
• February 14th 2011, 03:37 AM
Unknown008
Oh, then no, sorry, I don't know about Euclidean Geometry... I'd be most happy to help otherwise.
• February 14th 2011, 03:51 AM
Prove It
Quote:

Originally Posted by bram kierkels
Given a circle and a square next to the circle . How can I construct a rectangle with the same surface as the square in the circle? Only using a compass and ruler.

Thanks

I think the solution is that "squaring the circle" is impossible...
• February 14th 2011, 06:02 AM
bjhopper
the square is the maximum area quadrilateral which can be inscribed in a circle so the problem is impossible

bjh
• February 14th 2011, 06:23 AM
The original post doesn't give any values for the diameter of the circle
and the side of the square.
The circle certainly needs to be of a minimum size in relation to the square.

Also, may we assume "surface" refers to surface area ?
• February 14th 2011, 04:13 PM
bjhopper
construction of rectangle in circle
Followup to prior post

Given a square side lenght a and a circle diameter b.
the square when inscribed in a circle produces the largest area of the 4sided polygons.
A square with side a can be inscribed in a circle with diameter a*rad2. No rectangle with area a^2 can fit in this circle.An infinite number of smaller area rectangles will fit with areas ranging from close to 0 to a max close to a^2.These rectangles have diagonal lengths a*rad2

bjh