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in this video we're gonna focus on
calculating the area of a rhombus so
let's say if the diagonal that is the
distance between points B and C let's
say that diagonal is 15 units long and
the other diagonal between a and C let's
say it's 12 units long what is the area
of this particular promise the area of a
rhombus is 1/2 times the product of the
two diagonals so the first diagonal
let's call it 15 and the second one
let's call it 12 one half of 12 is 6 and
6 times 15 well 6 times 10 is 60 6 times
5 is 30 60 plus 30 is 90 so 6 times 15
is 90 and so that's how you can
calculate the area of a rhombus if
you're given the left of the two
diagonals now let's work on another
problem that's similar but also quite
different so once again let's say if we
have rhombus a b c d and also diagonal
AC and diagonal BD and they intersect at
Point E now the two diagonals they meet
at right angles now let's say if segment
B E is equal to 8 at segment C E is
equal to 10 what is the area of the
rhombus so this is 8 and this is 10 how
can we calculate the area now you need
to know that the diagonals of a rhombus
they're perpendicular bisectors of each
other they're perpendicular in the sense
that they meet our right angles and they
bisect each other so AE and EC are
congruent and B E is congruent to Eadie
so therefore that means that Eadie is 8
and AE is 10
so now we have the left of the to
diagnose so diagonal AC is ten plus ten
or twenty units long and diagonal BD is
eight plus eight or 16 is long so we
could use this formula again it's 1/2 D
1 times D 2 so D 1 is 16 D 2 is 20 half
of 16 is 8 and 8 times 20 if you have 8
$20 bills you have 160 bucks so this is
going to be 160 and so that's another
way in which you can calculate the area
of a rhombus if you're given those two
sides
is another problem for you so given the
same type of rhombus rhombus ABCD and
what's gonna say this is point e let's
say if we're told that CD is 17 units
long and AE is equal to 8 calculate the
area of the rhombus
so AE is 8 and CD is 17 how can we
calculate the area with this given
information you need to know that the
four sides of a rhombus are congruent so
therefore a B is 17 now we can calculate
B e because the rhombus can be broken up
into four couldn't grew a triangles so
let's focus on the right triangle the
hypotenuse is 17 and the base is 8 what
is the height how can we figure this out
so we need to use the Pythagorean
theorem see the hypotenuse is 17 a is 8
and let's calculate B 17 squared that's
289 8 times 8 is 64
now 289 minus 64 is 225 and so that's
equal to B squared and if we take the
square root of both sides the square
root of 225 is 15 so b e is 15 units
long
so now we can calculate the area so if B
E is 15 that means II D is 15 and if a E
is 8 E C is also 8 so this diagonal is
gonna be 8 plus 8 or 16 units long and
this one is 15 plus 15 or 30 units long
so the area is 1/2 d1 times d2 so that's
gonna be 1/2 times 30 times 16 so half
of 30 is 15 and 15 times 16 that's 240
and so that's the area of this
particular rhombus it's 240 square units
and the perimeter of a rhombus is 20 if
the length of one of the diagonals is
six what is the area of the rhombus so
let's draw a picture
so if we're given the perimeter what is
the side length of the rhombus keep in
mind that all four sides are congruent
so this is s then all sides are s so the
perimeter is simply four s so all we
need to do is divide the perimeter by
four and that will give us the side
length of the rhombus 20 divided by 4 is
5 so this is 5 units long now the length
of one of the diagonals is 6 so let's
say AC is 6 that means that a is 3 and
EC is also 3 so now we can calculate the
missing side of the right triangle we
can calculate B e so C squared is equal
to a squared plus B squared C the
hypotenuse is 5 a is 3 so let's
calculate B 5 squared is 25 3 times 3 is
9 and 25 minus 9 is 16 so 16 is equal to
B squared and now let's take the square
root of both sides so the square root of
16 is 4 so we have a 3 4 5 right
triangle so now we can calculate the
area so if B e is for e D is 4 which
means that diagonal BD is 8 so the area
is gonna be 1/2 D 1 times D 2 so that's
gonna be 1/2 times 8 times 6 now half of
a is 4 and 4 times 6 that's 24 and so
that's the area of this particular
rhombus it's 24 square units
now let's say this side is 30 and let's
say this part is 15 what's the area of
this rhombus so what do you think we
need to do now you need to know that a
rhombus is a type of parallelogram and
so a square is a parallelogram as well a
rectangle is a parallelogram and the
area of any type of parallelogram is the
base times the height in the case of a
rectangle the area is left times the
width but you can also treat this as
base times height you get the same thing
for a square the area is s squared or s
times F which is also based on site so
for any parallelogram whether it be it a
rectangle or square or rhombus the area
is based on site so the base of this
particular rhombus or parallelogram is
30 now we can see that the height is 15
so the area is just 30 times 15 and so
it's 450 square units
try this problem so let's say if we have
R on this ABCD and let's say that BC is
let me think about this let's say BC is
25 and BD is 24 actually let's make BD
14 so if BC is 25 and BD is 14 what is
the area of the rhombus
how can you calculate the area of this
figure well first what we need to do is
draw the other diagonal so we can't just
multiply base times side we don't know
the length of the height in this
particular case so we can't treat this
like a typical parallelogram
now if BC is 25 and if BD is 14 that
means B II has b7 and Edie has b7 as
well so now we just got to find a
missing side of this right triangle so
the hypotenuse is 25 the height of that
triangle 7 let's calculate the base of
the triangle so C squared is a squared
plus B squared C squared is 25 a is 7
and B is what we're trying to find that
25 times 25 is 625 7 times 7 is 49 and
so 625 minus 49 that's 576 so now let's
take the square root of both sides the
square root of 576 is 24 so C e is 24
and AE is also 24
so we can say that D 1 which is AC
that's 24 plus 24 or 48 and D 2 which is
BD that's 7 plus 7 or 14 so a is equal
to 1/2 D 1 times D 2 so that's 1/2 of 48
times 14 so 1/2 a 48 is 24 so now we
just gotta multiply 24 by 14 and so the
area of this particular rhombus is 336
square units so now you know how to
calculate the area of a rhombus
