A triangle has an area of $84{\text{cm}}^2$ and its three sides add up to 42 cm. What is the radius of its inscribed circle?

Join the incentre to the three vertices: the triangle splits into three smaller triangles, each with a side of the original as base and the inradius r as height. Adding them, the area equals ½r × (sum of the sides) = r × s, where s is the SEMI-perimeter. Here s = 42/2 = 21 cm, so r = 84/21 = 4 cm. (A 13-14-15 triangle fits the data exactly.) The trap 2 cm divides by the full perimeter instead of the semi-perimeter, 8 cm doubles the correct answer after making that same slip, and 21 cm reports the semi-perimeter itself.