The perimeter of an isosceles trapezoid circumscribed around a circle is $p$. Find the radius of this circle, given that the acute angle at the base of the trapezoid is $\alpha$.

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The lateral side of the given trapezoid is equal to a quarter of the perimeter.

## Solution

The lateral side of the trapezoid is $\frac{p}{4}$, the diameter of the inscribed circle is equal to the height of the trapezoid, i.e., $\frac{p}{4} \cdot \sin \alpha$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1b17058527caf5b5d058g-07.jpg?height=421&width=644&top_left_y=1516&top_left_x=707)

## Answer

$\frac{p}{8} \cdot \sin \alpha$