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in this video we're going to talk about
how to find the area
of an isosceles triangle
now keep in mind an isosceles triangle
has
two side lengths that are the same
they're of equal length
now in order to calculate the area
of any triangle
there's two things that we need to know
we need to know the base of the triangle
and the height of the triangle
if we could find those two pieces of of
information we could use this formula
area is one half base times height
now we have the base but we don't know
the value of the height so we got to
find that
what I'm going to do is I'm going to
break this triangle into two equal parts
but I'm only going to draw the left side
of that of what remains
so here it is
once you do that this becomes a right
triangle
and in this right triangle we have a
height of H
we only have one half the base now so
this is going to be eight
and this part is still 17.
because we have a right triangle
we could use the Pythagorean theorem to
calculate the value of H
so it's going to be a squared plus b
squared is equal to c squared
now let's make a equal to H
I mean a equal to 8 Rather B is going to
be the height h
the hypotenuse which is across the right
angle that's C that's going to be the
longest side of the triangle which is
17.
so now let's plug in what we have
8 squared plus
B is going to be H so this is H squared
and C is 17.
now 8 squared
8 squared is 8 times 8 which is 64.
17 squared I'm going to have to
use a calculator for that
17 times 17 is 289.
now to get H squared by itself
we need to subtract both sides by 64.
and we can bring down
the H squared
so 289 minus 64 is 200 and 225.
now to get H by itself we can take the
square root of both sides
and the square root of 225 is 15.
so the height of the triangle is 15
units
now let's redraw the original triangle
but only with the relevant parts that we
need
so the base is still 16.
but the height
we know the height to be
15.
this is the 8 15 17 triangle
so now we can calculate the area
now that we know the values of B and H
so it's one half times the base which is
16. times the height which is 15.
half of 16 is 8.
and 8 times 15
. if you want to do it mentally you can
write it this way
15 is 10 plus 5.
8 times 10 is 80.
8 times 5 is 40.
so this is going to give us 120.
so that is the area
of this particular isosceles triangle
it's 120 square units
now let's talk about another way in
which we can get the answer
anytime you know all three sides
of a triangle you could use huron's
formula
and here it is first
you need to calculate s which is a plus
b plus c
divided by 2.
now a b and c those are the three side
lengths of the triangle so we can say
that a is 16.
B is 17 and C is 17.
now once you have S you can calculate
the area using this formula it's the
square root of s
times s minus a times s minus B
times s minus C so what we're going to
do is we're going to use this formula or
these two formulas rather to confirm our
answer to make sure that this answer is
correct
so let's plug in the three side lamps
into the first equation
so it's going to be 16 plus 17 plus 17.
divided by 2.
16 plus 17.
that's 33
and 33
well let's do it this way let's add 17
plus 17 first
17 plus 17 is 34.
and 16 plus 34 is 50.
half of 50 is 25.
so our s value
is
25.
now that we know the value of x we can
use this formula to get the area
so we have the square root of 25
and then it's 25 minus a a is 16.
25 minus B we know B is 17 and so is C
now 25 minus 16 that's going to be 9.
25 minus 17 is 8.
and we have two of them
so now let's do some
simplifying
now I'm going to break this into three
parts first we have 25 which is a
perfect square so I'm going to leave it
by itself in a radical then we have the
square root of 9.
now we can't take the square root of
eight but 64 we can 8 times 8 is 64.
so this is what we now have
the square root of 25 is 5. the square
root of 9 is 3 and the square root of 64
is 8.
now 5 times 3 is 15. and we already know
what 15 times 8 is is the same as 8
times 15 which is 120.
so we do get the same answer
regardless of the methods that we choose
to use
so the area of this particular isosceles
triangle is 120 square units
now for the sake of practice
go ahead and try this problem
calculate the area of this isosceles
triangle feel free to pause the video
and work on this example
so we're going to use the first method
because it's a lot faster
so let's draw half
of this triangle
and let's calculate the height
so the base is going to be half of what
it was which is 7. but the hypotenuse is
still going to be 25.
now at this point it helps to know the
special right triangles
for instance there's the three four five
a right triangle
there's the 5 12 13 right triangle
there's the 8 15 17 triangle
and then there's the 724
25 triangle which is what we have so
knowing this you can automatically know
that H is 24. now there's some other
ones like the 940 41
and I believe 11 60 61. let me make sure
that's correct
and it is
for the most part
you'll have one of those six variants or
any multiple of them so for instance if
you multiply this by two you can have
the six eight
ten triangle or if you multiply it by
three then 9 12 15 triangle
so any multiples of those numbers will
work as well
so like this one if you multiply it by
two you have the 10 24 26 triangle
so now that we know the height of the
triangle
let's redraw the picture
so the base is 14.
and the height
is 24.
so we can go ahead and use this formula
the area is going to be one half base
times height
base is 14. and the height is 24.
one half of 14 is seven and so this
becomes 7 times 24.
so to do it mentally let's break down
24 into 20 plus 4.
7 times 20 if you have seven twenty
dollar bills how much money do you have
that's going to be 140.
and then 7 times 4 is 28. adding those
two together
we get 168.
so that's the area of the triangle
and let's confirm it real quick using
Heron's formula
so let's calculate s
is going to be one half the sum of the
three sides
so 14 plus 25 plus 25 divided by 2.
25 plus 25 is 50 50 plus 14 is 64.
half of 64 is 32. so we could say s is
32.
now the area is going to be the square
root of s
and then s minus a
let's say 32 minus 14 that's 18. and
then s minus B 32 minus 25 that's 7 and
then 32 minus the other 25 is 7 as well
32 we can write that as 16 times 2.
18 that's 2 times 9.
and 7 times 7 I'm going to leave it as
49. I'm trying to collect all the
perfect squares here
so 16 is the perfect square 2 times 2 is
4. we have 9 and 49.
the square root of 16 is 4 the square
root of 4 is 2 the square root of 9 is 3
and the square root of 49 is 7.
so 4 times 2 is 8.
and then 8 times 3 is 7 I mean no 8
times 3 is 24. but 24 times 7 that's
going to give us the same answer of 168
so we know we have the right answer
so that's it for this video now you know
two methods that you can employ in order
to calculate the area of an isosceles
triangle
