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πππ is an equilateral triangle with a side length of 92 centimeters.
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Three circular sectors are drawn in the triangle such that their centers are the vertices π, π, and π.
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The radius of each sector is 46 centimeters.
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Find the area of the part of the triangle bounded by the arcs of the circular sectors giving the answer to one decimal place.
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Letβs begin by sketching this out.
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Notice how the radius of each circle is 46 centimeters.
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That means that each sector must reach exactly halfway along the side of the triangle.
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The diagram will look something like this.
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Ad weβre trying to find the area of the shaded part.
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So what do we need to work out?
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Well, if we knew the area of the triangle and the area of the three individual sectors, we could find the difference between these.
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And it would tell us the size of the shaded area.
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So we recall that the area of a triangle could be found by using the formula, a half ππ sin π.
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And the area of a sector is a half π squared π for a circle with a radius π and an angle of π in radians.
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Remember, the interior angles of an equilateral triangle are each 60 degrees.
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And 60 degrees is the equivalent to a third π radians.
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So we can say that the area of the triangle is a half multiplied by 92 squared multiplied by sin of π over three.
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This is equal to 2116 root three.
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And then, we have three identical sectors.
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So we can find the area of them by multiplying the area of one of the sectors by three.
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Itβs three lots of a half multiplied by 46 squared multiplied by π over three.
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And this is equal to 1058π.
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The shaded area is the difference between these two values.
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Itβs 2116 root three minus 1058π, which is equal to 341.2144 and so on.
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We need to give our answer to one decimal place.
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The first digit after decimal point is two.
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And the deciding digit is a one.
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Remember, if the deciding digit is less than five, we round our number down.
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One is indeed less than five.
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So the shaded area correct to one decimal place is 341.2 centimeters squared.