Video: Finding the Area of a Regular Octagon Given the Length of Its Sides

The side of a regular octagon is 5 cm long. Find the area of the octagon giving the answer to two decimal places.

06:02

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.

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