We are told that triangle ABC has perimeter p and inradius r and then they want us to find the area of ABC in terms of p and r so we know that perimeters are just the sum of the sides of the triangle or how long of length will have to be if you wanna to go around the triangle and let's just remind ourselves what the inradius is if we take the angle bisectors of each of these vertexes each of these angles right over here so bisectors that right over there and then bisector that right over there this angle is going to be equal to that angle this angle is going to be equal to that angle and then this angle is going to be equal to that angle there and the point where those angle bisectors intersect that right over there is our incenter and it is equal distant from all of the three sides and the distance from those sides that's the inradius so let me draw the inradius so when you find the distance between the point and the line you wanna drop a perpendicular so this length right over here is the inradius this length right over here is the inradius and this length right over here is the inradius and if you want you could draw a incircle here with the center at the incenter and with the radius r and that circle would look something like this we don't have to actually draw for this problem so you could draw a circle that looks something like that then we call that the incircle so let's think about how we can find the area here especially in terms of this inradius well the cool thing about the inradius is this it looks like the altitude well this looks like the altitude for this triangle right over here triangle A let's label the center let's call it I for incenter so r is this r right over here is the altitude of triangle AlC this r is the altitude of triangle BlC and this r which we didn't label that r right over there is the altitude of triangle AlB and so we could find the area of each of those triangles in terms of both r and their basis maybe if we sum up the area of all the triangles we can get something in terms of our perimeter and our inradius so let's just try to do this so the area of the entire triangle the area of ABC is going to be equal to and I will color code this this is going to be equal to the area of AlC so that's what I am shading here in magenta is going to be euqal to the area of AlC plus the area of BlC which is this triangle right over here