Question: A right triangle has a hypotenuse of length $ z $ and an inscribed circle with radius $ c $. If the triangles legs are in the ratio $ 3:4 $, what is the ratio of the area of the circle to the area of the triangle? - Deep Underground Poetry

Understanding the Context

In today’s digital landscape, clean geometry and predictable proportions influence everything from mobile app layouts to data visualization dashboards. A right triangle with legs in a 3:4 ratio forms a well-known 3-4-5 triangle—commonly used in architectural scaling, 3D modeling, and responsive UI design. Adding an inscribed circle introduces depth in spatial reasoning: the circle touches all three sides, and its radius depends directly on the triangle’s shape.

As creators, educators, and tech practitioners seek reliable, consistent formulas, understanding the exact relationship between $ z $, $ c $, and the triangle’s geometry becomes essential—not only for academic clarity, but for building accurate tools, instructional content, and algorithmic models.

How the Triangle and Circle Interact – A Clear Breakdown

Let’s start with the triangle: legs in a 3:4 ratio, hypotenuse $ z $. Using the Pythagorean theorem:

Key Insights

Let the legs be $ 3k $ and $ 4k $. Then:

$ z^2 = (3k)^2 + (4k)^2 = 9k^2 + 16k^2 = 25k^2 \Rightarrow z = 5k $

So $ k = \frac{z}{5} $. This lets us express all triangle components in terms of $ z $.

The area $ A $ of the triangle is:

$ A = \frac{1}{2} \cdot 3k \cdot 4k = 6k^2 = 6\left(\frac{z^2}{25}\right) = \frac{6z^2}{25} $

Final Thoughts

Next, the radius $ c $ of the inscribed circle in any triangle is given by the formula:

$ c = \frac{A}{s} $, where $ s $ is the semi-perimeter.

Perimeter = $ 3k + 4k + z = 7k = \frac{7z}{5} $, so:

$ s = \frac{7z}{10} $

Then:

$ c = \frac{\frac{6z^2}{25}}{\frac{7z}{10}} = \frac{6z^2}{25} \cdot \frac{10}{7z} = \frac{60z}{175} = \frac{12z}{35} $