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Therefore, the ratio of the area of the circle to the area of the triangle is:
Question: An archaeologist discovers a triangular stone tablet with a right angle. The hypotenuse measures $ 13 $ cm, and the radius of the inscribed circle is $ 2 $ cm. What is the area of the triangle?
Using the formula for the inradius of a right triangle:
Question: A mammalogist observes that the cross-section of a whales tail is a circle inscribed in a right triangle with legs of lengths $ a $ and $ b $. If the hypotenuse is $ 10 $ cm and the inradius is $ 2 $ cm, find the area of the triangle.
Using the formula for area in terms of the inradius and semiperimeter:
Also, from the inradius formula for a right triangle:
Substitute into the area expression:
Question: An ornithologist tracks a birds flight path forming a triangle with side lengths 7 cm, 24 cm, and 25 cm. What is the radius of the inscribed circle in this triangle?
Question: A rectangle with dimensions 5 cm by 12 cm is inscribed in a circle. What is the circumference of the circle?
The diagonal of the rectangle is the diameter of the circle