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# The larger circle has center O and passes through D. The smaller circle has diameter OD. What percent of the larger circle's area is gray?

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The larger circle has center O and passes through D. The smaller circle has diameter OD. What percent of the larger circle's area is gray? Aug 18, 2017

#1
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 gray circle area = π * (radius)2 radius of the smaller circle  =  $$\frac{OD}{2}$$ gray circle area = π * ($$\frac{OD}{2}$$)2 gray circle area = π * $$\frac{OD^2}{4}$$ gray circle area = $$\frac{πOD^2}{4}$$

 larger circle area = π * (radius)2 radius of the larger circle  =  OD larger circle area = π * OD2

What, percent, is the gray circle's area out of the larger circle's area?

$$\frac{\text{gray circle area}}{\text{larger circle area}}\,=\,\frac{\frac{\pi OD^2}{4}}{\pi OD^2}\,=\,\frac{\pi OD^2}{4}\,*\,\frac{1}{\pi OD^2}\,=\,\frac14\,=\,\frac{25}{100}\,=\,25\%$$

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Aug 18, 2017

#1
+3
 gray circle area = π * (radius)2 radius of the smaller circle  =  $$\frac{OD}{2}$$ gray circle area = π * ($$\frac{OD}{2}$$)2 gray circle area = π * $$\frac{OD^2}{4}$$ gray circle area = $$\frac{πOD^2}{4}$$

 larger circle area = π * (radius)2 radius of the larger circle  =  OD larger circle area = π * OD2

What, percent, is the gray circle's area out of the larger circle's area?

$$\frac{\text{gray circle area}}{\text{larger circle area}}\,=\,\frac{\frac{\pi OD^2}{4}}{\pi OD^2}\,=\,\frac{\pi OD^2}{4}\,*\,\frac{1}{\pi OD^2}\,=\,\frac14\,=\,\frac{25}{100}\,=\,25\%$$

hectictar Aug 18, 2017
#2
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Thanks, hectictar......!!!!

Another thing to realize is that circles are to one another as the squares of their radiuses.....

Thus.....the smaller circle has a radius of 1/2 the larger circle...so...its area  = (1/2)^2  = 1/4  that of the larger circle   Aug 18, 2017