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The question is as follows:

Find the area of the shaded region in the terms of π. (No decimals)

diagram here

To figure this problem out, would I figure out what the area of a whole circle is and then somehow figure out what is missing from the circle and subtract so I only have the shaded region?

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Let the radius of the circle is $R$

thus you can see that $$(R/2)^2+(R/2)^2=14^2 $$ $$\frac{R^2}{2}=14^2 $$

Now you want to find the area of the shaded region which is nothing but $\theta \frac{R^2}{2}$

where $\theta=\frac{3\pi}{2}$ Thus the area of the shaded region is $$\frac{3\pi} {2}\cdot14^2 \hspace{5pt}cm^2$$

Hope it helps.

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  • $\begingroup$ This really helps! But I was just wondering if once we get to 3pi/2 • $14^2$ $cm^2$, should I simplify more? Like, multiply them together? I feel like if I do it would become messier but if I don’t it looks unfinished $\endgroup$ – Ella Aug 21 '18 at 22:28
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    $\begingroup$ yes, you can simplify it but let the pi be there don't put 3.14 or 22/7 $\endgroup$ – Deepesh Meena Aug 21 '18 at 23:49
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    $\begingroup$ $294 \pi$ is the answer $\endgroup$ – Deepesh Meena Aug 21 '18 at 23:52
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Calling the marked segments we can call $x$, it looks like the radius of the circle will be $2x$, the problem is much harder if it is not the radius. Since there is a right angle it sections off a quarter of the circle so the area of the shaded region will be $\pi(2x)^2-\frac{\pi(2x)^2}{4}$. Solving for $x$ comes from the Pythagorean theorem

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We can see that there is an isosceles right triangle with sides 14,r/2 and r/2 where r is the radius of the circle. Apply Pythagoras Theorem, 2•(r/2)²=14² Thus r²=2•14². Now the shaded area is 3/4th of the complete circle so the required area is: 3/4(pi)r²=3/4*pi*2*14²=294*pi.

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