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
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:
$\displaystyle l^2 + w^2 = (2r)^2$
Substitute l and w.
$\displaystyle (kx)^2 + \left(\dfrac{x}{k}\right)^2 = 4r^2$
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.
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
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