Because in any triangle, the area is $ A = r \cdot s $, where $ s $ is semiperimeter and $ r $ is inradius. This holds for right triangles as well.

["Why the Area of Any Triangle Equals $ A = r \cdot s $: The Universal Formula Explained", "In the world of geometry, one powerful and elegant principle stands out for all triangle types: the area $ A $ of a triangle can be expressed as $ A = r \cdot s $, where $ r $ is the inradius—the radius of the incircle that touches all three sides—and $ s $, the semiperimeter defined as half the sum of the triangle’s sides. This simple yet profound formula applies universally—whether the triangle is acute, obtuse, equilateral, or right-angled.", "### Understanding the Concept", "The formula $ A = r \cdot s $ reflects a deep geometric relationship between a triangle’s internal circle (incircle) and its overall shape. The semiperimeter $ s = \frac{a + b + c}{2} $, with $ a, b, c $ being the triangle’s side lengths, sets the stage. The inradius $ r $, the distance from the triangle’s incenter (center of the incircle) to any side, measures how “central” the circle is within the triangle. Multiplying $ r $ by $ s $ gives not just the area, but a powerful insight into triangle balance and symmetry.", "### Why It Holds for All Triangles", "This expression is not limited to right triangles—despite common misconception—and confirms its generality across all triangle types. Here’s why:", "- In right triangles, the inradius can be computed explicitly using $ r = \frac{a + b - c}{2} $, where $ c $ is the hypotenuse, and verified using $ A = \frac{1}{2}ab $ and computing $ s $. The equality holds as expected.\n- For general triangles, the formula follows from the fact that dividing the triangle into three smaller triangles via perpendiculars from the inradius center reveals that each contributes $ \frac{1}{2} r \cdot \ ext{side} $ to the total area. Summing these gives $ A = \frac{1}{2} r (a + b + c) = r \cdot s $.", "Thus, $ A = r \cdot s $ is a universal geometric identity rooted in the triangle’s internal symmetry and circular balance.", "### Practical Applications", "Knowing this formula offers clear benefits:", "- Quick area computation when the inradius and semiperimeter are known, avoiding cumbersome sine or height calculations.\n- Early identification of triangle type—if area and side measures yield $ A = r \cdot s $ consistently, verification strengthens confidence in classification.\n- Insight for educators and students: reinforcing the link between algebraic expressions and geometric properties enhances spatial reasoning and problem-solving skills.", "### Conclusion", "The relationship $ A = r \cdot s $ is more than a formula—it’s a window into the harmony of a triangle’s geometry. Whether exploring right triangles with intuitive clarity or general triangles with analytical depth, this identity unifies diverse shapes under a single, elegant principle. Embracing $ A = r \cdot s $ not only deepens geometric understanding but empowers smarter, faster calculations across all triangle forms.", "---", "Keywords: area of triangle, formula $ A = r \cdot s $, inradius, semiperimeter, right triangle area formula, universal geometry, triangle geometry, incircle radius, triangle area calculation"]