Question Video: Finding the Perimeter of the Decagon given Its Area Mathematics

The area of a regular decagon is 155.8 cm². What is the perimeter of the decagon?


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.

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