Video Transcript
The side of a regular octagon is
five centimeters long. Find the area of the octagon,
giving the answer to two decimal places.
In this question, we’re given some
information about a regular octagon. We’re told the length of its sides
are equal to five centimeters. We need to use this to find the
area of our octagon. And we need to give our answer to
two decimal places.
There’s actually two different ways
we can answer this question. The first way to answer this
question is to recall that we know a formula for finding the area of any regular
polygon. We recall the area of a regular
𝑛-sided polygon with side length 𝑥 is given by 𝑛𝑥 squared over four multiplied
by the cot of 180 divided by 𝑛 degrees. So one way of answering this
question will be to use this formula. We’re going to need to find the
number of sides of our polygon and the side length.
First, in the question, we’re told
that we’re dealing with a regular octagon. And we know an octagon has eight
sides, so our value of 𝑛 is going to be equal to eight. Next, we’re also told in the
question that the side lengths of our regular octagon are going to be equal to five
centimeters. So we’re going to set our value of
𝑥 equal to five centimeters. All we need to do now is substitute
these values into our formula for the area. Substituting 𝑛 is equal to eight
and 𝑥 is equal to five centimeters into our formula, we get the area of our regular
octagon is eight multiplied by five centimeters squared over four multiplied by the
cot of 180 divided by eight degrees.
And we can evaluate this
expression. First, eight multiplied by five
squared over four is equal to 50. Next, it’s also worth pointing out
we know we’re calculating an area and we can see our lengths are given in
centimeters. And in our formula, we get
centimeters squared. So our units are going to be
centimeters squared. Next, we can simplify our argument
for cot 180 divided by eight is equal to 45 over two. So this simplifies to give us the
cot of 45 over two degrees.
So we need to take the product of
these two terms. But remember, instead of
multiplying by the cotangent of an angle, we can instead divide by the tangent of
that angle. This gives us 50 divided by the tan
of 45 over two degrees centimeters squared. And we can just calculate this
expression. Well, we remember we need to have
our calculator set to degrees mode. We get 120.710 and this expansion
continues centimeters squared.
But remember, the question wants us
to give our answer to two decimal places. So we need to look at our third
decimal place to determine whether we need to round up or round down. Our third decimal place is
zero. And this is less than five. So this means we round down. And this gives us our final
answer. The area of a regular octagon with
side length five centimeters to two decimal places is 120.71 centimeters
squared.
Now we could stop here. However, we could also try and
answer this question without directly using our formula. To do this, we’ll start by
sketching our octagon. And it doesn’t need to be accurate
because this is just a sketch. All we need to know is that this is
a regular octagon. Next, we need to connect all eight
vertices of our octagon to the centre of our octagon, giving us the following eight
triangles. And it’s worth pointing out here,
we know that all eight of these triangles are congruent because they have exactly
the same lengths.
So, so far, we know that the area
of this octagon is going to be eight multiplied by the area of any one of these
triangles. So we want to find the area of one
of these triangles. And we’ll do this by splitting one
of them in half. This gives us the following
right-angled triangle. And it’s worth pointing out
here. 16 of these make up our
octagon. The base of this right-angled
triangle is half of one of the side lengths of our octagon. It’s going to be five over two
centimeters.
But to find the area of this
triangle, we want to know its height so we can use our formula a half the base
multiplied by the height. But there’s no easy way to directly
find the height of this triangle. So instead, we’re going to find the
interior angle of our triangle. And we can actually find this
directly from our diagram. 16 of these triangles make up our
regular octagon. That means all 16 of these
triangles in a row will make up a full turn. It’s 360 degrees. So this angle in our triangle is
360 divided by 16 degrees.
And if we were to simplify this, we
would get 45 over two degrees. Now, if we want to find the area of
this right-angled triangle, we’re going to want to find its height. We can see we know one of the
angles of this right-angled triangle and we know the opposite side length of this
triangle. So we’re going to do this by using
trigonometry. We recall if we have an angle 𝐴 in
a right-angled triangle, then the tan of 𝐴 is going to be equal to the length of
the side opposite angle 𝐴 divided by the length of the side adjacent to angle
𝐴.
Applying this to the angle we know
in our right-angled triangle, we get the tan of 45 over two degrees is equal to five
over two all divided by the height ℎ. We want to use this to find the
value of ℎ, so we’re going to need to multiply through by our value of ℎ and then
divide through by the tan of 45 over two degrees. And simplifying this, we get that ℎ
will be equal to five divided by two times the tan of 45 over two degrees
centimeters.
We’re now ready to find the area of
our octagon. First, the area of our right-angled
triangle is going to be equal to one-half times the base multiplied by the
height. This is one-half times five over
two centimeters multiplied by five over two tan of 45 over two degrees
centimeters. And finally, remember, 16 of these
right-angled triangles make up our regular octagon. So we can find its area by
multiplying this through by 16.
So all that’s left to do is
evaluate this expression. First, we have three factors of two
in our dominator. And we can cancel this with three
of the shared factors of two in 16. This just leaves us with a factor
of two in our numerator. And then we can simplify our
numerator. We have two multiplied by five
multiplied by five. And this is equal to 50. And we can also simplify our
units. We know this is an area. And we have centimeters multiplied
by centimeters. So this is going to be centimeters
squared.
So, in fact, this entire expression
simplifies to give us 50 divided by the tan of 45 over two degrees centimeters
squared. And this is exactly the expression
we already calculated. To two decimal places, this is
120.71 centimeters squared. Therefore, we were able to show two
different methods of calculating the area of a regular octagon with side length five
centimeters. In both cases, to two decimal
places, we got 120.71 centimeters squared.
