Inradius, perimeter, & area

we're told that triangle ABC has perimeter P and in radius R and then they want us to find the area of ABC in terms of P and R so we know that the perimeter is just the sum of the sides of the triangle or how long a fence would have to be if you wanted to go around the triangle and let's just remind ourselves what the in radius is if we take if we take the angle bisectors of each of these vertices each of these angles right over here so if I sect that right over there and then bisect 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 this angle is going to be equal to that angle there and the point where that ant those angle bisectors intersect that right over there is our in center and it is equidistant from all of the three sides and the distance from those sides that's the inner radius so let me draw the inner radius so when you find the distance between a point and a line you want to drop a perpendicular so this length right over here is the inner radius this length right over here is the inner radius and this length right over here is the in radius and if you want you could draw an in circle here that acts with the center at the in center and with radius R and that circle would look something like this we don't have to necessarily draw it for this problem so you could draw a circle that looks something like that and then we'd call that the in circle so let's think about how we can find the area here especially in terms of this in radius well the cool thing about the in radius is its it looks like the altitude or this looks like the altitude for this triangle right over here triangle a let's label the center let's call it I 4 in center so R is the this R right over here is the altitude of triangle AIC this R is the altitude of triangle B I see and this R which we didn't label that are right over there is the altitude of triangle a ib and we know and and so we could find the area of each of those triangles in terms of both r and their bases and maybe if we sum up the area of all the triangles we can get something in terms of our perimeter and our Radia 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'll color code this it's going to be equal to the area of a I see so that's what I'm shading here in magenta it's going to be equal to the area of a I see plus the area of B I see which is this triangle right over you actually do that in a different color I've already used the blue so let me do that in orange plus the area of B I see so that's this area right over here so plus the area of B I see and then finally plus the area I'll do this in a let's see I'll use this pink color plus the area of AIB plus the area of AIB that is the area a I be take the sum of the areas of these two triangles you've got the area of the larger triangle now AIC the area of AIC is going to be equal to 1/2 base times height so this is going to be one half the base is the length of AC one half AC times the height times this altitude right over here which is just going to be R times R that's the area of AIC and then the area of B I see bi C is going to be one half one half the base which is B C times the height which is R and then plus the area of AIB this one over here is going to be one half the base which is the length of this side a be a B times the height which is once again which is once again R and over here we can factor out a 1/2 R from all of these terms and you get 1/2 R times times AC plus BC plus BC plus a B and I think you see where this is going Plus that's a different shade of pink plus a B plus now what is AC plus BC plus a B AC plus BC plus a B well that's going to be the perimeter P that is the perimeter of P if you just take the sum of the sides so that is the perimeter of P and it looks like we're done the area of our triangle the area of our triangle of ABC is equal to 1/2 times R times the perimeter times the perimeter which is kind of a neat result 1/2 times the inner radius times the perimeter of the triangle or sometimes you'll see it written like this it's equal to R times P over s or sorry P over 2 P over 2 and this term right over here the perimeter divided by 2 is sometimes called the semi perimeter semi semi perimeter semi perimeter and sometimes it's denoted by s so sometimes you'll see the area is equal to R times s where s is the semi perimeter it's the perimeter divided by 2 I personally like it this way a little bit more because I remember that p is perimeter and this is useful because obviously now if someone gives you an in radius and a perimeter you can figure out the area of a triangle or if someone gives you the area of the triangle and the perimeter you can get B in radius so if you give you two of the of these variables you can always get the third so for example if someone if this was a triangle right over here but this may be the most famous of the right triangles if I have a triangle that has length three four and five we know this is a right triangle you can verify this from the Pythagorean theorem and if somewhere say what is the in radius of this triangle right over here well we can figure out the area pretty pretty easily we know this is a right triangle three squared plus four squared is equal to 5 squared so the area is going to be area is going to be equal to three times four times one half so three times four times one half is 6 and then the perimeter here is going to be equal to three plus four which is 7 plus 5 is 12 and so we have we have the area so let's write this area is equal to 1/2 the inradius times the perimeter so here we have 12 is equal to 1/2 times the in radius times the perimeter so we have oh sorry we have 6 let me write this in the area is 6 we have 6 is equal to 1/2 times the inner radius times 12 and so in this situation 1/2 times 12 is just 6 we have 6 is equal to 6 R divide both sides by 6 you get R is equal to 1 so if you were to draw the inner radius for this one which is kind of a neat result so let me draw some angle bisectors here draw some angle bisectors here this 3 4 5 right triangle has an in radius of 1 so this distance equals this distance which is equal this distance which is equal which is equal to which is equal to 1 which is kind of a neat result