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Find the area of the shaded part of the diagram in terms of π.
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To find the shaded area of this diagram, weβll need to find two areas.
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Itβs the area of the larger quarter circle minus the area of the smaller semicircle.
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And we know that the largest sector is indeed a quarter circle because we can see that this line is a tangent to the semicircle.
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And the angle between a tangent and a radius is 90 degrees.
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To calculate these areas, we recall the formula for area of a sector with radius π and angle π radians.
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Itβs a half π squared π.
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Remember, a full turn is two π radians.
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So a quarter of a turn, 90 degrees, must be two π divided by four or π over two radians.
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The radius of our larger quarter circle is 30 centimeters.
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So the area is a half multiplied by 30 squared multiplied by π over two.
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To multiply these three numbers, we add the denominator of one to 30 squared.
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And since 30 squared is 900, multiplying the numerators and we get 900π.
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Then, for the denominators, we get four.
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And so the area of a quarter circle is 900π over four square units.
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Then, for the semicircle, 180 degrees is a half of 360 degrees.
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And a half of two π is π radians.
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So 180 degrees is equal to π radians.
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This time, the area of this sector is one-half multiplied by 15 squared multiplied by π.
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15 squared is 225.
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So the area of our semicircle is 225π over two square units.
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To subtract these two areas, we could make the denominators the same.
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However, 900 over four and 225 over two are fairly easy to evaluate.
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900 divided by four is 225.
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And 225 divided by two is 112.5.
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And so the shaded area is the difference between these two numbers.
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Itβs 225π minus 112.5π.
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The area is therefore 112.5π square units.
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Now, in fact all units are centimeters.
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So we found that the area of the shaded region is 112.5π centimeters squared.