The product of the perimeter of a triangle, the radius of its in‐circle, and a number gives the area of the triangle. The number is

Concept:

• Incircle of a triangle is a circle which is inscribed by the triangle.
• Area of a triangle = \(\frac{1}{2}\) × base × height. 

Calculation:

Consider a triangle ABC with incircle with radius r.

∴ OF = OH = OG = r.

Consider Δ ABO.

Here OF is perpendicular to AB (AB is tangent to the circle)

Area of Δ ABO = \(\rm \frac{1}{2}× OF× AB\)

⇒ Area of Δ ABO = \(\rm \frac{1}{2}× r× AB\)

Similarly 

Area of Δ ACO = \(\rm \frac{1}{2}× OH× AC\)

⇒ Area of Δ ABO = \(\rm \frac{1}{2}× r× AC\)

Similarly 

Area of Δ OBC = \(\rm \frac{1}{2}× OG× BC\)

⇒ Area of Δ ABO = \(\rm \frac{1}{2}× r× BC\)

Area of Δ ABC = Area of Δ ABO + Area of Δ ACO + Area of Δ OBC

⇒ Area of Δ ABC = \(\rm \frac{1}{2}× r× AB\) + \(\rm \frac{1}{2}× r× AC\) + \(\rm \frac{1}{2}× r× BC\)

⇒ Area of Δ ABC = \(\rm \frac{1}{2}× r× \left(AB +BC+AC\right)\)

⇒ Area of Δ ABC = \(\rm \frac{1}{2}× r\) × Semi - perimeter.

Hence the number is 1/2