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The ratio of habitable volume to non-habitable volume is:
In a right triangle used to model seismic wave paths, the hypotenuse measures $d$ units, and the inradius of the triangle is $r$. If the wave reflects off the incircle center, forming a shortest path to a side, what is the ratio of the area of the incircle to the area of the triangle?
Let the right triangle have legs $a$, $b$, and hypotenuse $d$. The area $A$ of the triangle is:
The inradius $r$ of a right triangle is given by:
Also, the area can be expressed in terms of the inradius and semiperimeter $s =
But we want the ratio of the area of the incircle to the area of the triangle:
But we can express this ratio more elegantly using known identities. Alternatively, recall:
But instead of algebraic complexity, use a standard result: In a right triangle,
However, the cleanest route is to use known geometric identities.
We know $A = r(a + b - r)$, but more efficiently:
