Question Video: Finding the Area of the Shaded Part between a Circle and a Pentagon Mathematics

Find the area of the shaded region to the nearest tenth.

06:26

Video Transcript

Find the area of the shaded region to the nearest tenth.

In this question, we’re given a diagram and we need to determine the area of the region we’ve just shaded in this diagram, and we need to give our answer to the nearest tenth. To do this, let’s start by looking at our diagram. We can see we have a regular pentagon inscribed inside a circle. First, we know this is a regular pentagon because all of its side lengths are the same. And this guarantees all of its angles will be the same as well. Next, if we look at the area of the shaded region, we can see it’s the area between our pentagon and our circle. In other words, this is the area of the circle and then we remove the area of this pentagon. So the shaded area is going to be the area of our circle minus the area of our pentagon.

So to find the shaded area, we need to find two things. We need to find the area of the circle and we need to find the area of our regular pentagon. Let’s start by finding the area of our circle. Recall the area of a circle of radius 𝑟 will be 𝜋𝑟 squared. And in fact, we’re given the radius of our circle on the diagram. The radius of our circle is 49.5. Therefore, we can directly find the area of our circle. Its area is going to be 𝜋 multiplied by 49.5 all squared. And if we evaluate this, we get 2450.25 multiplied by 𝜋, and we could call this square units. However, since this is not technically necessary, we’ll leave this out for space, but it is worth keeping in mind.

Next, we’re going to want to find the area of our pentagon, and there are a few different ways of doing this. One way will be to use our formula for finding the area of any regular polygon. We recall the area of any regular 𝑛-sided polygon with side length 𝑥 is given by 𝑛 times 𝑥 squared over four multiplied by the cot of 180 over 𝑛 degrees. By using this formula, we can calculate the area of any regular 𝑛-sided polygon by finding the number of sides of our polygon and the length of one of its sides. We can see in our diagram our polygon has five sides; it’s a pentagon. So our value of 𝑛 is going to be equal to five. However, we’re not told the lengths of the sides of our pentagon, so we’re going to need to find these out.

Now, there are a few different ways we could do this. One way will be to add the following radius to our circle. Since this is the radius of a circle, we know that this length is going to be 49.5, and it’s not necessary to do this to answer our question. However, it will help us see what’s happening. We can do this for the rest of the vertices in our regular pentagon. Now we have five triangles, and all of these triangles have exactly the same length. These are five congruent triangles.

And the reason we’re pointing this out is we want to know the lengths of one of the sides of our pentagon. Right now, we’ve shown it’s in a triangle, and we know two of the lengths of our triangle. However, to find the value of 𝑥, we’re going to also need to know one of the angles in this triangle. And what we’ve just shown is at the center of our circle, we have five equal angles which add to give 360 degrees. So each of these angles is 360 divided by five degrees, and we can calculate this. It’s equal to 72 degrees.

So now we have a triangle where we know two of the sides and one of the angles and we need to find the length of the other side. And one way of finding this will be to use the law of cosines. We recall the law of cosines tells us if we have a triangle with side lengths lowercase case 𝑎, lowercase 𝑏, and lowercase 𝑐 and the angle opposite to side lowercase 𝑎 is given by capital 𝐴, then we know the following must be true. Lowercase 𝑎 squared is equal to lowercase 𝑏 squared plus lowercase 𝑐 squared minus two times lowercase 𝑏 times lowercase 𝑐 multiplied by the cos of capital 𝐴. Since we know the angle opposite to our side length 𝑥, we can use this to find an expression for 𝑥. We get that 𝑥 squared will be equal to 49.5 squared plus 49.5 squared minus two multiplied by 49.5 times 49.5 multiplied by the cos of 72 degrees.

We can then calculate the expression on the right-hand side of this equation. Remember, our calculator will need to be set to degrees mode. We get 𝑥 squared will be equal to 3386.162 and this expansion continues. Then we can find the value of 𝑥 by taking the square root of both sides. Remember, our value of 𝑥 is a length, so this must be positive. Taking the square root of both sides, we get 𝑥 is equal to 58.190 and this expansion continues.

And it’s worth pointing out here since 𝑥 represents a length, we could call this length units. However, it’s not necessary to answer this question. There is one more thing worth pointing out though. It might be very tempting to round our value of 𝑥 at this point. However, we shouldn’t do this yet. We should always round our answer right at the very end of the question. Otherwise, we might end up with the wrong answer. So it’d probably be very useful to add the exact value of 𝑥 to our calculator’s memory.

Now that we know both the value of 𝑛 and the value of 𝑥, we can substitute these into our formula for the area of our pentagon. Substituting these values in, we get the area of our pentagon is equal to five multiplied by 58.190 and this continues squared all over four multiplied by the cot of 180 divided by five degrees. And we can simplify this expression. First, 180 divided by five is equal to 36. Next, we recall multiplying by the cotangent of an angle is the same as dividing by the tangent of that angle. So we can instead divide by the tan of 36 degrees giving us the following expression for the area of our regular pentagon. And if we calculate this expression, we get 5825.815 and this continues square units. And once again, it’s important not to round here because we need to round our value at the end. So it might be worth putting this into our calculator’s memory.

Now that we found both the area of our circle and the area of our pentagon, we can find the area of the shaded region. The area of the shaded region is the area of the circle minus the area of our pentagon. We’ve calculated these; it’s equal to 2450.25𝜋 minus 5825.815 and this continues. And if we calculate this, we get 1871.871 and this continues square units. But remember, the question only wants us to give our answer to the nearest tenth. So we need to look at our second decimal place to determine whether we need to round up or round down. We can see the second decimal place in this expansion is seven, and this is greater than or equal to five. So this tells us we need to round up, giving us 1871.9 square units, which is our final answer.

Therefore, by using our formula for calculating the area of a circle, our formula for calculating the area of a polygon, and the law of cosines, we were able to find the area of the shaded region given to us in the question to the nearest tenth. We got that this was 1871.9 square units.

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