# Math Help - Construction Rectangle in Circle

1. ## 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

2. 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.

3. Thanks Unknown008. But I ment a ruler without numbers. So this should be a question in the topic "Euclidean Geometry". Can you help?

4. Oh, then no, sorry, I don't know about Euclidean Geometry... I'd be most happy to help otherwise.

5. 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...

6. the square is the maximum area quadrilateral which can be inscribed in a circle so the problem is impossible

bjh

7. 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 ?

8. ## 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