The shown figure represents a circle inside a regular hexagon. Find the area of the shaded regions, giving your answer to the nearest tenth.
In this question, we’re given a figure. And in this figure, we have a circle which is inside of a regular hexagon. We need to use our figure to determine the area of all of the shaded regions added together. And we need to give our answer to the nearest tenth.
So to answer this question, let’s start by looking at the shaded area on our figure. We can see the shaded area is part of the area between our hexagon and our circle. We would know how to calculate all of the area between our circle and our hexagon. We would need to find the area of our hexagon and then subtract the area of the circle. But then, because this is a regular hexagon and our circle is inscribed inside of our hexagon, each of these segments is going to be equal in area. So because three of the six areas are shaded in, we need to multiply this by one-half. Therefore, the area of the shaded region is one-half times the area of the hexagon minus the area of the circle.
So to find the area of our shaded region, we need to find two things. We need to find the area of the hexagon and the area of our circle. Let’s start by finding the area of our hexagon. There’s a few different ways we could do this. We’re going to use the formula for finding the area of a 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 to find the area of our hexagon, we’re going to need to find the value of 𝑛 and the value of 𝑥. Of course, the value of 𝑛 is the number of sides, and any hexagon has six sides. So our value of 𝑛 is six. However, we’re not given any of the side lengths of our hexagon in our diagram. So we’re going to need to find these. And there’s a few different ways we can do this. The easiest way to do this is to know that the internal angle in a regular hexagon is 120 degrees. If we then split this in half with the following line, we get something interesting.
Since we split an angle of 120 degrees in half, the angle here is 60 degrees. The same is true on the other side. We’re splitting the internal angle of a regular hexagon. The angle here is also 60 degrees. Therefore, all of the internal angles in this triangle are 60 degrees. This must be an equilateral triangle. And in an equilateral triangle, all of the lengths are the same. So our side length of our hexagon must be 14. Therefore, in our formula for the area of a hexagon, our value of 𝑥 is 14.
And it’s worth pointing out though this wasn’t the only way we could’ve calculated this value. If we did not know the internal angle of a regular hexagon was 120 degrees, we could construct the same triangle and we would know two of the lengths were 14. Then we can construct the following six congruent triangles. Then we can see we’re splitting an angle of 360 degrees into six equal segments. So we can find the angle between our two lengths of 14 to be 60 degrees.
So in this triangle, we would then know two lengths and the angle between them. We can then use the law of cosines to find the other length of our triangle. This would also give us the answer of 14. However, this method is more complicated.
We’re now ready to find the area of our hexagon. Our value of 𝑛 is six, and our value of 𝑥 is 14. So we substitute these in. The area of our hexagon is six times 14 squared over four multiplied by the cot of 180 divided by six degrees. And we can evaluate this. First, six times 14 squared over four is equal to 294, and 180 divided by six is 30. So this simplifies to give us 294 multiplied by the cot of 30 degrees. And to evaluate this, we need to recall multiplying by the cotangent of an angle is the same as dividing by the tangent of that angle. So we can simplify this to give us 294 divided by the tan of 30 degrees.
But 30 degrees is one of our standard angles. We know the tan of 30 degrees will be one divided by the square root of three. And of course, dividing by one over square root three is the same as multiplying by the square root of three. So we can actually find the area of our hexagon exactly. It’s 294 root three square units.
Now, to find the area of our shaded region, we’re going to need to find the area of our circle. And to find this, we recall the area of a circle of radius 𝑟 is given by 𝜋𝑟 squared. So to find the area of our circle, we’re going to need to find the length of its radius. The easiest way to do this is to draw the following radius onto our diagram. This is called the apothem, and it has several very useful properties.
First, the angle it makes with the sides of our hexagon will be right angles. It will be 90 degrees. Next, it will bisect the angle at the center of our circle. Since this was 60 degrees, we have 60 over two is 30 degrees. And although it’s not necessary to answer this question, the apothem will also bisect the sides of our hexagon. We want to use this information to find the value of our radius 𝑟.
We can see that 𝑟 is in a right-angled triangle. We know the hypotenuse of our right-angled triangle, and we know the angle between the hypotenuse and 𝑟. So we can find the value of 𝑟 by using trigonometry. We know the cosine of an angle in a right-angled triangle is equal to the length of the side adjacent to this angle divided by the length of our hypotenuse. So in our triangle, the cos of 30 degrees is equal to 𝑟 divided by 14. We can find the value of 𝑟 by multiplying both sides of this equation through by 14. And once again, 30 degrees is a standard angle. So we know the cos of 30 degrees is the square root of three divided by two. So we can find the value of 𝑟 exactly. 𝑟 is 14 multiplied by root three over two. And 14 over two is seven. So 𝑟 is equal to seven root three.
Now that we’ve found the length of the radius of our circle, we can use this to find the area of our circle. It’s equal to 𝜋𝑟 squared, which in this case is 𝜋 times seven root three all squared. Once again, we can find the area of the circle exactly. We’ll start by distributing the square over our parentheses. This gives us 𝜋 times seven squared multiplied by root three squared. We can simplify further. Seven squared is 49, and root three all squared is equal to three. So we get 𝜋 times 49 times three. And 49 times three is equal to 147. So the area of our circle is 147𝜋 square units.
Now that we’ve found both the area of our hexagon and our circle, we’re able to find the area of the shaded region. Remember, this is one-half multiplied by the area of the hexagon minus the area of the circle. Substituting in our expressions for the area of the hexagon and the area of the circle, we get one-half multiplied by 294 root three minus 147𝜋. And if we calculate this, we get 23.704 and this expansion continues square units.
But remember, the question wants us to give our answer to the nearest tenth. So we’re going to need to round this. Nearest tenth is to the nearest one decimal place. We need to check our second decimal place. The second decimal place in this expansion is zero. So we’re going to need to round down. This gives us 23.7. And although it’s not necessary because this represents an area, it’s useful to give this a unit. We’ll write these as square units.
Therefore, in this question, we were able to find the shaded region, which was part of a region between a circle inscribed inside of a regular hexagon. To do this, we needed to use trigonometry and our formula for finding the area of a regular hexagon and the area of a circle. We were able to show the area of this region to the nearest tenth is 23.7 square units.