Find the area of the shaded part in the figure. Round your answer to the nearest tenth.
Remember, the area of a sector with a radius 𝑟 and an angle 𝜃 radians can be found by using the formula a half 𝑟 squared 𝜃. In this case, we have two sectors that are shaded. In fact though, these sectors are identical. And that’s because vertically opposite angles are equal. This means the angle of each sector must be equal. And therefore, we can find the area of one of these sectors and then multiply it by two to find the area of the shaded part.
To find the angle of one of these sectors, we recall that angles on a straight line sum to 180 degrees. Since we know that the angle given is 114 degrees, we can find the angle of one of our sectors by subtracting 114 from 180. That’s 66 degrees. We’re not quite ready to use this in our formula yet. Remember, we said that the formula only worked when the angle was measured in radians. So let’s find a way to convert from degrees to radians.
Two 𝜋 radians is equal to 360 degrees. We can find the amount of radians that are equal to one degree by dividing this equation by 360. And if we do, we see that one degree is equal to 𝜋 over 180 radians. To find the angle of each sector in radians, we multiply its angle in degrees by 𝜋 over 180. And when we do, we can see that 𝜃, the angle of our sector, is 11𝜋 over 30 radians.
The radius of our circle is six. So the area of one of the sectors is a half multiplied by six squared multiplied by its angle in radians, which we now know is 11𝜋 over 30. And if we type that into our calculator, we get 33𝜋 over five as the area of one of the sectors.
To ensure we have the most accurate answer, we’re going to leave this in terms of 𝜋 for a moment. And we’re going to multiply it by two to find the total of the shaded area. That’s 66𝜋 over five. And if we now convert this into decimal form, we can see that the shaded areas 41.469 and so on metres squared.
We were told to round our answer to the nearest tenth though. That’s the first decimal place. The number in the tenths column is four. And the digit immediately to its right is the deciding digit. Remember, if that deciding digit is five or above, we round the number up. If it’s less than five, we round the number down. Six is greater than five. So we round the number up. This tells us that 41.469 and so on is closer to 41.5 than it is to 41.4.
And we see the shaded area is 41.5 meters squared.