Video Transcript
The area of a regular decagon is
155.8 centimetres squared. What is the perimeter of the
decagon?
Now first of all, a decagon
remember has ten sides. So we’re interested in a polygon
where 𝑛 is equal to 10. Now looking at the question, we’re
not given the side length anywhere. But we want to work out the
perimeter. So this is a question about working
backwards from knowing the area, deducing the side length. And then using that to calculate
the perimeter of the decagon. So we’re going to need our area
formula. So let’s recall that from the
previous work. So there it is. Area is 𝑛𝑥 squared over four
multiplied by cot 180 over 𝑛. Now we know that 𝑛 is equal to 10
because we’re working with a decagon. And we also know that this area is
equal to 155.8 centimetres squared. So we can set ourselves up an
equation. And here it is. 10 lots of 𝑥 squared over four
times cot of 180 over 10 is equal to 155.8.
So now we have an equation that we
can solve in order to work out the value of 𝑥. 𝑥 remember represents the side
length. And once we have that, we can then
easily work out the perimeter. So let’s just simplify this
equation a bit first of all. 10 over four will simplify to five
over two. And also cot of 180 over ten. Well that’s just cot of
eighteen. So we actually have this equation
here. Now we want to get 𝑥 squared on
its own. So we got that factor of two in the
denominator. We’ve also got a factor of tan 18
in the denominator because remember cot is one over tan. So I’m gonna multiply both sides of
the equation by two tan 18. And that would give me that five 𝑥
squared is equal to two tan 18 multiplied by 155.8. Then I can divide both sides of the
equation by five, which will give me this line here.
And I’ll evaluate that on my
calculator. Remember I’m working in degrees
here. So that gives me 𝑥 squared is
equal to 20.24899 and so on. Next we need to take the square
root in order to work out what 𝑥 is. So I use my calculator to evaluate
the square root of 20.248. And that gives me 𝑥 is equal to
4.499888. And this value to any set of
reasonable degree of accuracy is 𝑥 is equal to 4.5. So I’ve worked out the side length
of one side of this decagon.
But I haven’t finished the question
because the question asked me to calculate the perimeter. Now for the perimeter I just need
to add up all of the sides. Well there are ten of them. And they’re all the same
length. Therefore, the perimeter is gonna
be 10𝑥. So ten lots of this value we’ve
just worked out of 4.5. So the perimeter of this decagon is
45 centimetres.
