Inradius Perimeter and Area

We're told the 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 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 inradius is.

If we take the angle
bisectors of each of these

vertices-- each of these
angles right over here.

So bisect 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 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 equidistant 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 a point and a line,

you want to 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 an 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 necessarily
draw it for this problem.

So you could draw a circle
that looks something like that.

And then we'd 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 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 for incenter.

This r right over here is
the altitude of triangle AIC.

This r is the altitude
of triangle BIC.

And this r, which we didn't
label, that r right over there

is the altitude of triangle AIB.

And we know-- 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 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'll

color code this-- is going to
be equal to the area of AIC.

So that's what I'm
shading here in magenta.

It's going to be equal to
the area of AIC plus the area

of BIC, which is this
triangle right over here.

Actually let me do that
in a different color.

I've already used the blue.

So let me do that in orange.

Plus the area of BIC.

So that's this area
right over here.

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.

That is the area AIB.

Take the sum of the areas
of these two triangles,

you 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 1/2.

The base is the length
of AC, 1/2 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 BIC is
going to be 1/2 the base,

which is BC, times the
height, which is r.

And then plus the area of
AIB, this one over here,

is going to be 1/2
the base, which

is the length of this side
AB, times the height, 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
AC plus BC plus AB.

And I think you see
where this is going.

Plus-- now that's a different
shade of pink-- plus AB.

Now what is AC plus BC plus AB?

Well that's going to
be the perimeter, 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 ABC
is equal to 1/2 times r times

the perimeter, which is
kind of a neat result.

1/2 times the inradius times
the perimeter of the triangle.

Or sometimes you'll see
it written like this.

It's equal to r times P
over s-- sorry, P over 2.

And this term right over here,
the perimeter divided by 2,

is sometimes called
the semiperimeter.

And sometimes it's denoted
by s so sometimes you'll

see the area is
equal to r times s,

where s is the semiperimeter.

It's the perimeter divided by 2.

I personally like it
this way a little bit

more because I remember
that P is perimeter.

This is useful because obviously
now if someone gives you

an inradius 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 the inradius.

So if they give you
two of these variables,

you can always get the third.

So for example, if this was
a triangle right over here,

this is maybe the most famous
of the right triangles.

If I have a triangle that
has lengths 3, 4, and 5,

we know this is
a right triangle.

You can verify this from
the Pythagorean theorem.

And if someone
were to say what is

the inradius of this
triangle right over here?

Well we can figure out
the area pretty easily.

We know this is
a right triangle.

3 squared plus 4 squared
is equal to 5 squared.

So the area is going to be
equal to 3 times 4 times 1/2.

So 3 times 4 times
1/2 is 6 and then

the perimeter here
is going to be

equal to 3 plus 4, which
is 7, plus 5 is 12.

And so we have the area.

So let's write this area is
equal to 1/2 times the inradius

times the perimeter.

So here we have 12 is equal
to 1/2 times the inradius

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 inradius times 12.

And so in this situation,
1/2 times 12 is just 6.

We have 6 is equal to 6r.

Divide both sides by 6,
you get r is equal to 1.

So if you were to draw
the inradius for this one,

which is kind of a neat result.

So let me draw some
angle bisectors here.

This 3, 4, 5 right triangle
has an inradius of 1.

So this distance
equals this distance,

which is equal to
this distance, which

is equal to 1, which is
kind of a neat result.