Video Transcript
Find the total area of the shaded regions in the regular polygons below, giving your answer to the nearest 10th.
In this question, we’re asked to find the total area of the shaded region given in the diagram below. And in this diagram below, we’re told that all of our shapes are regular polygons. And we need to give our answer to the nearest 10th. To answer this question, let’s start by looking at our diagram to take a look at the shaded region. The first thing we can notice because all of the shapes in our diagram are regular polygons is that the shaded region lies entirely within a regular hexagon with side length 39.
However, we can also see some area is cut out. For example, we can see that the area of this triangle is taken out of the shaded region. And because we’re told that these are regular polygons and we also can see one of the side lengths of this triangle is 39, we know this is an equilateral triangle with side length 39. So, if we found the area of the regular hexagon and then removed the area of this equilateral triangle, we would be closer to finding the area of our shaded region.
However, we can see from the sketch we still have a problem. There’s still more area we need to remove. We’re going to need to remove the area between the square and the pentagon. Once again, both of these are regular polygons, and we know their side lengths are all going to be 39. So, by using all of this information, we actually have two different equivalent ways of finding the shaded area.
The first method would be to find the area of our regular hexagon, subtract the area of our regular pentagon, add the area of our square, and then subtract the area of our equilateral triangle. However, it might be easier to instead do this in steps. Let’s start by finding the area of the outer region. The area of the outer shaded region is just going to be the area of the hexagon. And then we subtract the area of the pentagon.
By using similar logic, we can find the area of our inner region; we’ll mark this 𝐼. To find the area of our inner region 𝐼, we take the area of the square and then we subtract the area of our equilateral triangle. It doesn’t matter which of these two methods you would prefer. Both of them will give the same answer. We’re going to use the second method.
To use either method, the first thing we’re going to need to do is find the area of all four of our regular polygons. And although we could do this directly by using geometry, it’s far easier to just recall the formula. We know the area of any regular 𝑛-sided polygon of side length 𝑥 is given by 𝑛𝑥 squared over four multiplied by the cot of 180 divided by 𝑛 degrees. So, we’ll start by using this formula to find the area of our hexagon.
We know that hexagons have six sides. So, our value of 𝑛 is equal to six. And in our diagram, we can see the side length is 39. So, our value of 𝑥 is 39. So, by substituting 𝑛 is equal to six and 𝑥 is equal to 39 into our formula, we get the area of our hexagon is six times 39 squared over four multiplied by the cot of 180 divided by six degrees. And it is possible to evaluate this expression without a calculator because 180 divided by six is 30. And we know multiplying by the cotangent of an angle is the same as dividing by the tangent of that angle. And we know the tan of 30 degrees is one divided by the square root of three.
However, it’s not possible to find the exact area of all of our shapes without using a calculator. So, we can just use this for this example as well. The only thing we will do is rewrite our formula. So instead of multiplying by the cot of 30 degrees, we’re dividing by the tan of 30 degrees. And if we calculate this expression, we get 3951.673 and this expansion continues.
And at this point, we might be tempted to round our answer. However, it’s very important we don’t round until the very end of our question; otherwise, we might get the incorrect answer. So, it’s important we either put this number into our calculator’s memory or we remember the exact expression we used to find it. Also, because this number represents an area, we could give this the units of square units. But to save space, we’re not going to include this; however, it’s always worth keeping this in mind.
Now that we found the area of our hexagon, we’re going to do exactly the same to find the area of our pentagon. From our diagram, we know that our pentagon is a regular pentagon. And we know all of its side lengths will be 39. So, we substitute 𝑛 is equal to five and 𝑥 is equal to 39 into our formula. We get the area of our pentagon is five times 39 squared over four multiplied by the cot of 180 degrees over five.
And we’re going to calculate this in exactly the same way we did before. 180 over five is 36. And instead of multiplying by the cot of 36 degrees, we’ll divide by the tan of 36 degrees. This gives us five times 39 squared over four multiplied by the tan of 36 degrees. And if we calculate this expression, we get 2616.846 and this expansion continues square units. And once again, it’s important we don’t round this value because we need to round our values at the end.
Now that we found the area of our hexagon and the area of our pentagon, we can find the outer shaded region that we’ve labeled 𝑅. We just subtract the area of our pentagon from the area of our hexagon. And by using the exact values of the area of our hexagon and pentagon, we get that 𝑅 is 1334.827 and this expansion continues square units.
We’re going to follow the exact same process to find the area of the inner region 𝐼. We might be tempted to use our formula to find the area of our square. However, we know the area of a square is just the square of one of its sides. So, in this case, the area of our square is just going to be 39 squared. And we can calculate this; it gives us 1521 square units. And we could use our formula, and it would give us the correct answer. However, this method is easier.
Now, we want to find the area of our triangle. And there’s a few different options for doing this. For example, because this is an equilateral triangle, we know all of its interior angles are 60 degrees. So, we could find the area of this triangle by finding the height of this triangle by using trigonometry and then using half the base multiplied by the height. However, we can also just use our formula. In this case, we’ll use our formula. Our value of 𝑛 is going to be three, and our value of 𝑥 is 39.
So, the area of our triangle is three times 39 squared over four multiplied by the cot of 180 over three degrees. And we’ll calculate this in the same way we did before. 180 over three is 60 degrees. And instead of multiplying by the cot of 60 degrees, we’re going to divide by the tan of 60 degrees. And because we know the tan of 60 degrees is the square root of three, this is another area which we can find an exact value for. The area of our triangle is three times 39 squared over four multiplied by the tan of 60 degrees. And we could give an exact form for the area of this triangle. However, it’s not necessary; we’ll just write out its decimal expansion. It’s equal to 658.612 and this expansion continues square units.
We can now find the area of the interior shaded region in our diagram. It’s going to be the area of the square minus the area of our triangle. And if we calculate this using the exact values, we get 862.387 and this expansion continues square units.
Now we’re finally ready to find the entire area of the shaded region. It’s going to be the area of the outer region added to the area of our inner region, 𝑅 plus 𝐼. And once again, if we calculate this by using the exact values, we get 2197.215 and this expansion continues square units.
But remember, the question wants us to give our answer to the nearest 10th. This means to one decimal place. So, we need to determine whether we need to round up or round down. To do this, we need to look at our second decimal place. This is equal to one, which is less than five. So, we’re going to round down. And this gives us our final answer of 2197.2 square units.
In this question, we were able to use our formula for finding the area of regular polygons to find the area of a complicated shaded region given to us in the diagram. To the nearest 10th, we were able to show that this was equal to 2197.2 square units.
