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
This proof explores the value of the radius R in a configuration of a circle and rectangle inside a square. Follow the steps below to see how R is calculated geometrically.
First, we identify the known sides of the square and rectangle. Next, we determine the relationship between the circle and rectangle. Finally, we calculate the radius R using geometric properties.
To solve the problem, we simplify it by adding an angle \(x\) like it is shown in the next figure.
Now, let’s calculate:
\[
\cos x = \frac{R-1}{R}, \quad
\sin x = \frac{R-2}{R}
\]
\[
(\sin x)^2 + (\cos x)^2
= \left(\frac{R-2}{R}\right)^2 + \left(\frac{R-1}{R}\right)^2
= 1
\]
\[
\frac{R^2 – 4R + 4 + R^2 – 2R + 1}{R^2} = 1
\]
\[
2R^2 – 6R + 5 = R^2 \quad \Rightarrow \quad R^2 – 6R + 5 = 0
\]
\[
(R-1)(R-5) = 0
\]
The solution \(R=1\) is not accepted because \(\cos x \neq 0\). Therefore:
\[
\color{orange}{R=5}
\]
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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)