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SSAT Upper Level
Quantitative (Math)

Area of Isosceles Triangle in Regular Decagon

Very Hard Geometry And Measurement Polygons

Consider a regular decagon (10-sided polygon) inscribed in a circle. This decagon has been divided into 10 congruent isosceles triangles by drawing lines from the center of the circle to each vertex of the decagon.

If the radius of the circumscribed circle is 12 cm, what is the area of one of these isosceles triangles?

Use the formula for the area of an isosceles triangle: $$A = \f\frac{1}{2} \cdot b \cdot h$$, where $$b$$ is the base and $$h$$ is the height. The base of each isosceles triangle is one side of the decagon, and can be calculated using the formula: $$b = 2 \cdot r \cdot \sin\left(\f\frac{\pi}{n}\right)$$, where $$n$$ is the number of sides, and $$r$$ is the radius of the circumscribed circle.

Your task is to find the area of one of these triangles.

Hint

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