Here is an equilateral triangle and
a regular pentagon. What is the size of the angle
marked 𝑎? Show all your working.
Well, the first thing we can do is
actually find out what the angles are, so the interior angles of the equilateral
triangle. And the way we do that is by
actually dividing 180 by three. And that’s because each of the
angles in an equilateral triangle are the same. And what I’ve done here is I’ve
actually shown the reason why, so make sure at every stage you actually show why and
the reason why is because it’s the interior angle of an equilateral triangle.
So now the next thing we want to do
is actually find out what the interior angle of our regular pentagon is. And the reason we want to do this
is because if we find this, then this plus 60 degrees plus 𝑎 is gonna be equal to
360, because their angles are on a point and therefore allows us to solve the
problem and find 𝑎.
There’re actually a couple of
methods we can actually use to find the interior angle of the regular pentagon. So we’re just gonna show a couple
here. The first one is using this
formula, which is that the interior angle is equal to 180 multiplied by 𝑛 minus two
over 𝑛, where 𝑛 is actually the number of sides of the regular polygon. And the key here is the fact that
we have a regular polygon, and in this case it’s a regular pentagon. And that means that we know we can
use this formula because it will only work if we do have a regular polygon.
So therefore, if we actually
substitute our values in, we’re gonna have the interior angle of the pentagon, which
is 𝐼 𝑝 — I’ve wrote it here — is equal to 180 multiplied by five minus two over
five. And we use five because that’s the
number of sides. Well, this is gonna give us 180
multiplied by three over five, which will give us 540 over five, which will give us
an interior angle of 108 degrees, because if we do 540 divided by five, we get
So as I said, we’re actually gonna
check by using another method. And in this other method, we’re
gonna have to use another formula and a relationship. And that formula is that the
exterior angle of a polygon is equal to 360 divided by 𝑛, where 𝑛 again is the
number of sides. And then the relationship we’re
gonna use is that an interior angle plus an exterior angle is equal to 180
So first of all, what we’re gonna
do is actually work out the exterior angle of our pentagon. And we can do this using 360
divided by five, cause five is the number of sides. I’ve actually marked onto the
diagram where an external angle would be. And if we divide 360 by five, we
get 72 degrees. So we can see that, actually, yes,
our exterior angle is gonna be equal to 72 degrees.
So now what we’re gonna use is
actually our other relationship we know, which is that an interior angle plus an
exterior angle is equal to 180 degrees. So we’ve got our 𝐼 𝑝, which is
our interior angle of the pentagon, plus 72, because that’s our exterior angle, is
equal to 180. So therefore, the interior angle of
the pentagon is gonna be equal to 180 minus 72, because we’ve actually subtracted 72
from each side. And therefore, what we get is an
answer of 108 degrees.
And great, yet this is actually the
same interior angle that we got using the other method. So we know, yes, we definitely know
that the interior angle of our pentagon is 108 degrees. So now what we’re gonna do is
actually work out our angle 𝑎. And as we said earlier, what we
need to do that is actually use the fact that angles at a point are equal to 360
degrees. So therefore, 60 plus 108 plus 𝑎
is equal to 360. So then we have 168 plus 𝑎 is
equal to 360. So then if we actually subtract 168
from each side, we get 𝑎 is equal to 360 minus 168.
So therefore, we can say that our
final answer is that 𝑎 is equal to 192 degrees. And I’ve just drawn that again on
the diagram. And we’ve used at this point the
relationship that angles at a point add up to 360 degrees.