What expression represents the area of the square inscribed inside a circle with radius r?
You need to find the expression for the area of the square inscribed in a circle in terms of the radius of the circle given as r.
A square inscribed in a circle is one that has the largest sides and can be fitted into the circle, because if that were not the case the corners of the square would not touch the circle.
There are two properties that you may have learnt, one of them is that a right triangle inscribed by a circle has the hypotenuse as a diameter. If you haven't learnt that, we can start with the fact that the length of the longest line segment which can be drawn in a circle is equal to its diameter. Now, the square is divided by its diagonal into two congruent right triangles each with two sides equal to the sides of the square and the hypotenuse equal to the diagonal of the square. The diagonal has the same length as the diameter of the circle.
The radius of the circle is r. The diagonal of the square is equal in length to 2r. If the side of the square is s, use the Pythagorean Theorem to get the length of the diagonal as sqrt( s^2 + s^2)
or sqrt (2*s^2) = s*sqrt 2
s*sqrt 2 = 2r
=> r = s*sqrt 2/ 2
=> r = s/sqrt 2
s = r* sqrt 2
The area of a square with side s is equal to s^2.
As s = r*sqrt 2
=> s^2 = r^2*( sqrt 2)^2
=> s^2 = 2*r^2
Hope you understood how I got the result.
So we get the area of the square inscribed by a circle in terms of the radius of the circle as 2*r^2.
DRAW YOUR PICTURE with inscribed square and have scissors handy! Draw diagonals which will cross at the circle's centerpoint and label each line to the corners with r.
The square INSIDE the circle makes the DIAGONAL of that square the length of 2r. Cut the OTHER diagonal and you will have 4 triangles with two sides of r length. The third and longest side will be the chord (line segment where corners touch the the sides of the circle) and if you paste quickly enough you can ignore its length :-)
Rearranging the four triangles, so that the chords become the adjoining sides you can make these four triangles into two squares of area r^2.
THEN, You can SEE the two squares so:
Area of inscribed square is 2r^2.
Once they let me cut loose in geometry, I became MUCH better!