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Find the area of the shaded part in the figure.
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Round your answer to the nearest tenth.
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Remember, the area of a sector with a radius ๐ and an angle ๐ radians can be found by using the formula a half ๐ squared ๐.
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In this case, we have two sectors that are shaded.
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In fact though, these sectors are identical.
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And thatโs because vertically opposite angles are equal.
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This means the angle of each sector must be equal.
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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.
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To find the angle of one of these sectors, we recall that angles on a straight line sum to 180 degrees.
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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.
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Thatโs 66 degrees.
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Weโre not quite ready to use this in our formula yet.
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Remember, we said that the formula only worked when the angle was measured in radians.
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So letโs find a way to convert from degrees to radians.
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Two ๐ radians is equal to 360 degrees.
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We can find the amount of radians that are equal to one degree by dividing this equation by 360.
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And if we do, we see that one degree is equal to ๐ over 180 radians.
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To find the angle of each sector in radians, we multiply its angle in degrees by ๐ over 180.
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And when we do, we can see that ๐, the angle of our sector, is 11๐ over 30 radians.
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The radius of our circle is six.
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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 11๐ over 30.
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And if we type that into our calculator, we get 33๐ over five as the area of one of the sectors.
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To ensure we have the most accurate answer, weโre going to leave this in terms of ๐ for a moment.
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And weโre going to multiply it by two to find the total of the shaded area.
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Thatโs 66๐ over five.
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And if we now convert this into decimal form, we can see that the shaded areas 41.469 and so on metres squared.
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We were told to round our answer to the nearest tenth though.
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Thatโs the first decimal place.
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The number in the tenths column is four.
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And the digit immediately to its right is the deciding digit.
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Remember, if that deciding digit is five or above, we round the number up.
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If itโs less than five, we round the number down.
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Six is greater than five.
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So we round the number up.
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This tells us that 41.469 and so on is closer to 41.5 than it is to 41.4.
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And we see the shaded area is 41.5 meters squared.