Question Video: Finding the Area of the Shaded Part between Two Concentric Circles

In this figure, the diameter of the larger circle is 41 cm and both circles have the same center. Determine, to the nearest tenth, the area of the shaded part.

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Video Transcript

In this figure, the diameter of the larger circle is 41 centimetres and both circles have the same center. Determine, to the nearest tenth, the area of the shaded part.

A good way to think about this question is as two individual sectors. We have a larger sector shown, from which the smaller sector has been removed. And by doing so, we’re left with the shaded area. To find the size of the shaded area then, we’re going to subtract the area of the smaller sector from the area of the larger sector.

We’ll begin by calculating the area of the larger sector. The formula we need is a half multiplied by 𝑟 squared multiplied by 𝜃. And that’s for an area of a sector with radius 𝑟 and angle 𝜃 radians. There are a couple of things we’ll need to consider before we can use this to calculate the area of either sector.

Firstly, we’ve been given the measure of the obtuse angle, where as our sector has a reflex angle. We recall that angles around a point sum to 360 degrees. So, we subtract 134 from 360 to get the measure of the reflex angle. That’s 226 degrees.

Next, we’ll need to convert this angle to radians. To do this, we recall that two 𝜋 is equal to 360 degrees. We’re going to scale down to find the amount of radians in one degree. And to do that, we divide by 360. Two 𝜋 over 360 simplifies to 𝜋 over 180. So, one degree is equal to 𝜋 over 180 radians. We can convert 226 degrees into radians by multiplying by 𝜋 over 180. And when we do, we can see the size of the angle 𝜃 in our sector is 113𝜋 over 90.

We now know the angle of each sector. We just need to find the radius. The diameter of the larger circle is 41 centimetres. We can find its radius by halving that. Half of 41 is 20.5. So, the area of the large sector is a half multiplied by 20.5 squared multiplied by 113𝜋 over 90. That gives us 828.826 and so on centimetres squared. We’re not going to round this just yet. Instead, we’ll repeat the process for the smaller sector, and then use the exact value in our later calculations.

This time, the angle 𝜃 is the same but the radius is going to need to be worked out. The radius of the larger circle was 20.5, so we need to subtract 12.3 from 20.5. And when we do, we see that the radius of the smaller circle is 8.2. And we can use this to find the area of the sector. It’s a half multiplied by 8.2 squared multiplied by 113𝜋 over 90. That’s 132.612 and so on centimetres squared.

We can subtract the area of the smaller sector from the area of the larger sector. And when we do, we get 696.213 and so on. We were told to give our answer to the nearest tenth. That’s the first decimal place. The first number after the decimal point is two. The digit immediately to its right is called the deciding digit. Remember, if that deciding digit is less than five, we round our number down. If it’s five or above, we round the number up.

Here, one is less than five, so we round our number down. And it tells us that this is closer to 696.2 than it is to 696.3. And the area of the shaded region is 696.2 centimetres squared.

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