Two overlapping circles with equal radius form a shaded region as shown in the figure. Express the area of the region and the complete perimeter (combined arc length) in terms of r and the distance between center, $D$ $D$ ? Let $r = 4 and D = 6$ $r=4 and D=6$ and calculate?

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Two overlapping circles with equal radius form a shaded region as shown in the figure. Express the area of the region and the complete perimeter (combined arc length) in terms of r and the distance between center, $D$ $D$ ? Let $r = 4 and D = 6$ $r=4 and D=6$ and calculate?

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Answer 1 · 2016-10-29T05:18:54

see explanation.

Explanation:

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Given $A B = D = 6, \Rightarrow A G = \frac{D}{2} = 3$ $AB=D=6, => AG=D/2=3$
Given $r = 3$ $r=3$
$\Rightarrow h = \sqrt{r^{2} - {(\frac{D}{2})}^{2}} = \sqrt{16 - 9} = \sqrt{7}$ $=> h=sqrt(r^2-(D/2)^2)=sqrt(16-9)=sqrt7$
$sin x = \frac{h}{r} = \frac{\sqrt{7}}{4}$ $sinx =h/r=sqrt7/4$
$\Rightarrow x = {41.41}^{\circ}$ $=> x=41.41^@$

Area GEF (red area) $= π r^{2} \cdot (\frac{41.41}{360}) - \frac{1}{2} \cdot 3 \cdot \sqrt{7}$ $=pir^2*(41.41/360)-1/2*3*sqrt7$
$= π \cdot 4^{2} \cdot (\frac{41.41}{360}) - \frac{1}{2} \cdot 3 \cdot \sqrt{7} = 1.8133$ $=pi*4^2*(41.41/360)-1/2*3*sqrt7=1.8133$

Yellow Area $= 4 \cdot$ $= 4 *$ Red Area $= 4 \cdot 1.8133 = 7.2532$ $= 4*1.8133=7.2532$

arc perimeter $(C \to E \to C) = 4 \times 2 π r \times (\frac{41.41}{360})$ $(C->E->C)=4xx2pirxx(41.41/360)$
$= 4 \times 2 π \times 4 \times (\frac{41.41}{360}) = 11.5638$ $= 4xx2pixx4xx(41.41/360)=11.5638$

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Two overlapping circles with equal radius form a shaded region as shown in the figure. Express the area of the region and the complete perimeter (combined arc length) in terms of r and the distance between center, $D$ $D$ ? Let $r = 4 and D = 6$ $r=4 and D=6$ and calculate?

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Two overlapping circles with equal radius form a shaded region as shown in the figure. Express the area of the region and the complete perimeter (combined arc length) in terms of r and the distance between center, DD? Let r=4andD=6r=4 and D=6 and calculate?

1 Answer

Explanation:

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Two overlapping circles with equal radius form a shaded region as shown in the figure. Express the area of the region and the complete perimeter (combined arc length) in terms of r and the distance between center, $D$ $D$ ? Let $r = 4 and D = 6$ $r=4 and D=6$ and calculate?