hey this is presh talwalkar and you're
watching mind your decisions
the william lowell putnam mathematical
competition
or more commonly called the putnam
competition
is an annual exam for undergraduate
college students in the united states
and canada
you have three hours to solve six
problems and then another three hours to
solve another six problems
that's a total of six hours for 12
questions or an average of 30 minutes
per problem
each question is worth 10 points for a
maximum of 120 points
but most people don't do that well
the average median score is about one
point
even though the students taking it are
generally really really good at math
most of the participants have spent
hours and hours preparing for the exam
and they're willing to take an extra six
hour test on the first saturday in
december
around the time of their other college
final exams so they're generally very
dedicated students who take this exam
in this video we'll focus on the 1978
putnam exam problem b1
a convex octagon inscribed in a circle
has four consecutive sides of length
three and four consecutive sides of
length two
find the area of the octagon
can you figure it out
give this problem a try and when you're
ready keep watching the video for a
solution
[Music]
so let's get started by sketching the
shape
we have a circle
and then we have four consecutive sides
of length three
and four consecutive sides of length two
i'll admit this is about as far as i got
with this problem
because i could not figure out a way to
deconstruct the shape into areas that i
could solve
however
i like learning about these kinds of
problems so that i can share interesting
problem solving techniques with you
and this particular problem has a very
elegant solution
the creative solution depends on slicing
up the shape like a pizza
so we take the center of the circle
and then we slice it up into triangles
and each of these lengths is the radius
of the circle
so we now have a shape that consists of
four isosceles triangles with side
lengths r r and three and four isosceles
triangles with side lengths r r and two
the insight is that we can rearrange
these triangles to create a new octagon
with exactly the same area as the
original octagon
so how can we do that
we'll take these green triangles and
form across
and then we'll fill in the remaining
areas with these yellow triangles
we now have an octagon with alternating
side lengths of 3 and 2.
so now how can we figure out the area of
this shape
well again you need to think outside the
box
we'll create a square around this
octagon
now each of these corners is an
isosceles right triangle
so each length here is the square root
of two
the rest of the lengths are three
so we can calculate the area of the
octagon is equal to the area of the
square minus the area of four of these
triangles
we can calculate each of these areas
pretty routinely
the area of the square will be the area
of a square with a side length of three
plus two times the square root of two
so we square the side length and we get
seventeen plus twelve times the square
root of two
we then calculate the area of one
triangle as one half times the square
root of two times the square root of two
which equals one
so the area the octagon will equal the
area the square minus four times the
area of a triangle and this simplifies
to be thirteen plus twelve times the
square root of two
and that's the answer
did you figure it out
thanks for watching this video these
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