𝐴𝐵, 𝐵𝐶, and 𝐶𝐷 are sides of a
regular 10-sided polygon. And 𝐶𝐷 and 𝐶𝐸 are sides of a
regular octagon. Work out the size of angle
𝐶𝐸𝐵. You must show all of your working
First of all, we’re actually gonna
mark on to the diagram some of the things that we know. First of all, because we actually
have 𝐶𝐷 as a shared side — so it’s a side of the regular 10-sided polygon and it’s
also a side of the regular octagon — so therefore we can actually say that the sides
𝐵𝐶 and 𝐶𝐸 must also be the same, because we’ve already said, like I said, that
𝐶𝐷 is a shared side in both of the polygons.
Okay, great! because of this, we
actually know that the triangle 𝐵𝐶𝐸 is actually going to be an isosceles
triangle. And this is gonna be really helpful
a bit later on in the question. To enable us to actually find the
angle 𝐶𝐸𝐵, which is what the question is looking for, we need to first of all
find out the angle 𝐵𝐶𝐸. And the way we can actually do that
is by splitting it in two, because actually what we’ve got here are two exterior
angles. So what we actually have is the
exterior angle of our 10-sided polygon, which is called a decagon, and also the
exterior angle of the regular octagon, so our eight-sided shape.
Okay, so how do we find out what
the exterior angle is going to be? Well, there are actually a couple
of ways we can solve this using different relationships and formula. So first of all, we know that the
exterior angle of a regular polygon is equal to 360 degrees divided by 𝑛, where 𝑛
is the number of sides of the polygon.
The key word that allows us to use
this is actually regular because we do know that the exterior angle of any polygon
add up to 360 degrees. However, we won’t be able to
calculate the individual values of those exterior angles if it wasn’t a regular
polygon. But because it’s a regular polygon,
we can actually do that, because we can just divide 360 degrees by 𝑛, the number of
Okay, so that’s one method. The next method is actually to
calculate one of the interior angles. And you can calculate an interior
angle by 180 multiplied by 𝑛 minus two over 𝑛. And that’s where 𝑛 again is the
size of our polygon. And we can use that to solve this
kind of problem, because actually we know another relationship and that’s the
exterior plus interior angles add up to 180 degrees. And this is because their angles
are on a straight line.
Okay, great! So we’ve got some different ways to
solve this. First of all, I’m gonna start using
the first formula to solve it. So I’m gonna start with the
exterior angle of our decagon. And I’ve actually used 𝐸𝐷 to
represent that. So we can say that 𝐸𝐷 is equal to
360 divided by 10, because it’s 10 sides, which gives us an angle of 36 degrees.
And now we’re gonna move on to the
exterior angle of the octagon. And to calculate the exterior angle
of our octagon, it’s gonna be 360 degrees divided by eight, which gives an angle of
Now as I said before, we can
actually check this by using the other method. So if we do, we’ve got the interior
angle of 𝐷, so the interior angle of our decagon, is equal to 180 multiplied by 10
minus two over 10, which is equal to 1440 divided by 10, which gives us 144
degrees. So that’s our interior angle.
So therefore, using our final
relationship that an exterior plus an interior angle equals 180 degrees, we can
state that the exterior angle of the decagon is gonna be equal to 180 minus 144,
which gives us 36 degrees.
So first of all, we can say that
the interior angle of our octagon is equal to 180 multiplied by eight minus two
divided by eight, which gives us 1080 divided by eight, which is equal to 135
degrees. Okay, great! So again now we can move on and
find the exterior angle of the octagon. And again, we use our final
relationship that exterior plus interior equals 180 degrees. So we can say that the exterior
angle of our octagon is equal to 180 minus 135, which gives us an angle of 45
degrees, which again, great, fits the angle that we got using the first method.
Okay, so now let’s move on and
actually work out the size of angle 𝐶𝐸𝐵. So what I’ve actually done is drawn
a sketch of the triangle that we’re using here. So we’re trying to find the angle
𝐶𝐸𝐵. But we’ve just found out the angle
𝐵𝐶𝐸, because the angle 𝐵𝐶𝐸 is equal to 36 degrees plus 45 degrees, cause they
are exterior angles that we calculated. So this is gonna give us an angle
of 81 degrees.
Okay, great! So now what’s the next step? Well, the next step is to actually
use the relationship that we found very early on. And that’s that this is actually an
isosceles triangle. So therefore, 𝐶𝐸𝐵 and 𝐶𝐵𝐸 are
actually the same. So they’re equal to each other. So therefore, we can say that
𝐶𝐸𝐵 plus 𝐶𝐵𝐸 is gonna be equal to 180 minus 81, cause that’s just the angles
in a triangle adding up to 180 degrees. And this gives an answer of 99
So great! We know that if we add those two
angles together, it’s gonna give us 99 degrees. So therefore, angle 𝐶𝐸𝐵 is gonna
be equal to 99 divided by two. That’s because as it’s an isosceles
triangle, we’ve already said 𝐶𝐸𝐵 and 𝐶𝐵𝐸 are gonna be the same. So then we get our final
answer. And that’s that 𝐶𝐸𝐵 is equal to
49.5 degrees, cause 99 divided by two gives us 49.5 degrees.