#### 800Question: A product manager designs a triangular logo with vertices at $(0, 0)$, $(a, 0)$, and $(0, b)$. The inradius of the triangle is $r$. Express the ratio of the area of the inscribed circle to the area of the triangle in terms of $r$ and the triangles semi-perimeter $s$.

Fri, 17 Apr 2026 08:50:34 +0000

#### 800Question: A product manager designs a triangular logo with vertices at $(0, 0)$, $(a, 0)$, and $(0, b)$. The inradius of the triangle is $r$. Express the ratio of the area of the inscribed circle to the area of the triangle in terms of $r$ and the triangle’s semi-perimeter $s$. Curious about how geometric precision shapes brand identity? This question reflects a growing interest among designers and product teams in quantifying simplicity through mathematical relationships—especially in minimalist, symmetric shapes like right triangles. With increasing focus on brand equity in the digital space, understanding spatial and proportional language like inradius and semi-perimeter offers a new lens for strategic design decisions. Right now, this concept is gaining traction across design and UX communities in the United States, driven by demand for mathematically grounded visual identity systems. A right triangle with legs along the axes creates a clean, scalable form ideal for logo use. With vertices at $(0, 0)$, $(a, 0)$, and $(0, b)$, the triangle’s area is $\frac{1}{2}ab$. The semi-perimeter $s = \frac{a + b + c}{2}$, where $c = \sqrt{a^2 + b^2}$ is the hypotenuse. The inradius $r$ is known