Circumscribed Circle Radius of a Pentagon

Hard Geometry Polygons

In a regular pentagon, each side has a length of $s$. The pentagon is inscribed in a circle, which means that all its vertices touch the circle. You are to calculate the radius $R$ of the circumscribed circle in terms of the side length $s$.

The formula that relates the side length $s$ of a regular polygon with $n$ sides to the radius $R$ of the circumscribed circle is given by:

$$ R =\frac{s}{2 imes ext{sin}\frac{ heta}{2}} $$

where $$ heta =\frac{360^ ext{o}}{n} $$ is the central angle subtended by each side. For a pentagon, $n = 5$. Calculate the radius $R$ of the circumscribed circle.

$R = \frac{s}{2 \times \text{sin}(36^\text{o})}$

$R = \frac{s}{2 \times \text{sin}(72^\text{o})}$

$R = \frac{s \cdot 2}{\text{sin}(36^\text{o})}$

$R = \frac{s \cdot 2}{\text{sin}(72^\text{o})}$

Hint

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