## What is the value of the circle's radius \(R\) ? (circle and rectangle in square )

Home -> Solved problems -> Calculate the radius R (circle and rectangle in square)

### Solution

To solve the problem, we simplify it by adding an angle \(x\) like it is shown in the next figure.

Now, let’s calculate \[\cos x=\frac{R-1}{R}\] \[\sin x=\frac{R-2}{R}\] \[\left(\sin x\right)^{2}+\left(\cos x\right)^{2} =\left(\frac{R-2}{R}\right)^{2}+\left(\frac{R-1}{R}\right)^{2}=1 \] \[\left(\frac{R-2}{R}\right)^{2}+\left(\frac{R-1}{R}\right)^{2}=1 \] \[\frac{R^{2}-4R+4+R^{2}-2R+1}{R^{2}}=1 \] \[2R^{2}-6R+5=R^{2}\] \[R^{2}-6R+5=0\] \[\left(R-1\right)\left(R-5\right)=0\] The solution \(R=1\) is not accepted because \(\cos x\neq0\), Therefore \[\color{orange} {R=5}\]

Home -> Solved problems -> Calculate the radius R (circle and rectangle in square)

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Home -> Solved problems -> Calculate the radius R (circle and rectangle in square)