Video: Determining the Area of a Regular Hexagon given Its Circumcircle Equation

Determine, to the nearest hundredth, the area of a regular hexagon whose circumcircle is 𝑥² + 𝑦² − 14𝑥 − 12𝑦 + 9 = 0.

07:06

Video Transcript

Determine, to the nearest hundredth, the area of a regular hexagon whose circumcircle is 𝑥 squared plus 𝑦 squared minus 14𝑥 minus 12𝑦 plus nine is equal to zero.

In this question, we’re tasked with finding the area of a regular hexagon. We need to find this area to the nearest hundredth. And to do this, we’re told that this hexagon has a circumcircle given by the following equation. To answer this question, let’s start by sketching the information we’re given. First, we’re dealing with a regular hexagon and we’re told that this hexagon has the circumcircle. And remember, the circumcircle of any polygon will touch all of the vertices of a polygon. And because we’re dealing with a regular polygon, we know the center of the circle and the center of this polygon will be the same.

The next thing we can do is use the equation of our circle to find the center of our circle and its radius. To do this, we’re going to need to write it in the general form for the equation of a circle. To do this, let’s start by rearranging our equations so we have the 𝑥-terms first and then the 𝑦-terms. And to write this in the general equation of a circle, we’re going to need to complete the square twice, once for our 𝑥-terms and once for our 𝑦-terms.

Let’s start with the 𝑥-terms. We need to take half the coefficient of our 𝑥. We get 𝑥 minus seven all squared, but then we’re adding an extra constant of 49. So we need to subtract 49. And to see this, we can distribute the square over our parentheses and then simplify. We get 𝑥 squared minus 14𝑥. We’re then going to want to do the same for our 𝑦-terms. We take half the coefficient of 𝑦 which is negative six, giving us one minus six all squared. But then we’re adding an extra 36. So we need to subtract this. And once again, if we distribute the square over our parentheses and then simplify these two terms, we’ll get 𝑦 squared minus 12𝑦. And then in our equation, don’t forget, we need to add nine, and this is equal to zero.

This is now almost in the general form for the equation of a circle. We just need to write all of our constants on the other side of our equation. We add 49 to both sides of our equation, add 36 to both sides for our equation, and subtract nine from both sides of our equation. We get 𝑥 minus seven all squared plus 𝑦 minus six all squared is equal to 49 plus 36 minus nine. And if we calculate the right-hand side of our equation, we see it’s equal to 76. And now we’ve written our equation in the general form for the equation of a circle. We know if the equation of a circle was given in the form 𝑥 minus 𝑎 all squared plus 𝑦 minus 𝑏 all squared is equal to 𝑟 squared, the center of our circle will be the point 𝑎, 𝑏 and the radius of our circle will be 𝑟.

In our case, we can find the coordinates of the center. That’s going to be the point seven, six. However, it won’t be necessary to answer this question. What we will need to notice is the radius of our circle is going to be the square root of 76. Now that we have the radius of our circle, let’s see if we can use this to find the area of our hexagon. And in fact, there’s a lot of different methods we could use to answer this question. We’ll only go through a few of these. The easiest way to answer this question is to know that the internal angle of a regular hexagon is equal to 120 degrees. This then gives us a really useful result when we add the following two radii to our diagram.

Since these two lengths are radii, we know that length is going to be the radius of our circle, which we know is the square root of 76. We’ll just call this 𝑟. Secondly, because these lengths go from the vertices of our hexagon to the center of our circle, they’re going to cut the angle in half. These internal angles of our triangle are 60 degrees. Of course, if two angles in our triangle are 60 degrees, the other angle also has to be 60 degrees. This is an equilateral triangle. And of course, in an equilateral triangle, all of the sides have the same length, so the side length of our hexagon is the square root of 76.

Now, one way of finding the area of our hexagon would be to use the formula for finding the area of a regular polygon. And this would work. We know this is a regular polygon, we know it has six sides, and we know its side lengths are root 76. However, this is not the only way we could find this area. By adding the following radii, we can turn our hexagon into six equilateral triangles. And in fact, these are congruent equilateral triangles. So we only need to find the area of one of these triangles and then multiply it by six to find the area of our hexagon. And we have a lot of different methods we could use to find the area of these equilateral triangles.

For example, we know all of the side lengths so we could use Heron’s formula. However, the easiest method would be to use half the base multiplied by the height. To do this, let’s start by sketching one of our equilateral triangles. To find this area, we’re going to need to know the height of this equilateral triangle. So we’ll do this by creating the following right-angled triangle. There’s a few different ways of finding the height of this triangle. One way is to use trigonometry. We know the sine of the angle in a right-angled triangle is equal to the length of the side opposite that triangle divided by the length of the hypotenuse.

Applying this to the angle 60 degrees in our right-angled triangle, we get the sin of 60 degrees is equal to ℎ divided by root 76. And we can then solve this for ℎ by multiplying through by root 76 and recalling the sin of 60 degrees is equal to root three over two. So ℎ is root 76 multiplied by root three over two. And if we calculate this, we see that ℎ is equal to root 57. Now we can find the area of one of these equilateral triangles by using half the base multiplied by the height. This gives us one-half times root 76 times root 57. But remember, the area we’re interested in is the area of our hexagon. And this is six of these equilateral triangles. So we need to multiply this by six.

And we can calculate this exactly. It’s equal to 114 root three. But the question doesn’t want us to give our answer exactly. It wants us to give it to the nearest hundredth, and this is the same to two decimal places. So if we find this for our answer, we get 197.45. And because this represents an area, we’ll call these square units. But this wasn’t the only way of answering this question. We can actually answer this question without knowing the internal angle of a regular hexagon. So let’s see how we would do this.

First we could construct the same six triangles we did before. We can still see that these triangles are congruent because they all share the same lengths. And we can notice something interesting about the angles at the center of our hexagon. These angles are all the same and they add to 360 degrees. So each of these are 60 degrees. So we can still find one angle in our triangles. We know that this will be 60 degrees. And don’t forget, we do know the radius of our circle is the square root of 76. Now there are two options. We can notice that because this is an isosceles triangle, the two angles at the base of our triangle have to be equal. And we can calculate these angles. They’ll both be 60 degrees.

Or alternatively, we can use the fact that we have the length of two sides in the angle to find the length of the other side by using the law of cosines. If we did this, we would also see that it’s the square root of 76. And then once again, we have all three side lengths are the same. So these must be equilateral triangles. Then once again, we have the same choices as before. We can find the area of each of these triangles and then multiply this by six to find the area of our hexagon. Or we can use our formula for finding the area of our regular hexagon since we know the side lengths and we know that a regular hexagon has six sides. Either way, we’ll get the same answer.

In this question, we were able to show, to the nearest hundredth, the area of the regular hexagon with circumcircle with equation 𝑥 squared plus 𝑦 squared minus 14𝑥 minus 12𝑦 plus nine is equal to zero is given by 197.45 square units.

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