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Challenging "Find the Hashed Area" Question

See attached question. It asks: "A circle is inside a square, which is also found inside another square with with 6 meter sides. Find the hashed area".

Is the correct answer: 9pi/2 meters^2 or 9pi meters^2?

Can you please systematically outline the steps that you took to arrive at the answer?

Thank you for your help and explanation.

Re: Challenging "Find the Hashed Area" Question

The side length of the smaller square is $\displaystyle 3 \sqrt{2}$ (do you see why?).

Therefore the diameter of the circle is $\displaystyle 3 \sqrt{2}$, so the radius is $\displaystyle \frac{3 \sqrt{2}}{2}$. The area of the circle is $\displaystyle (\frac{3 \sqrt{2}}{2})^2 \pi = \frac{9 \pi}{2}$.

Re: Challenging "Find the Hashed Area" Question

Hi Richard,

Thanks for the quick response. I do not see how you got the side length of the smaller square to be http://latex.codecogs.com/png.latex?3%20\sqrt{2}.

Can you please explain how you got to that step?

Thanks a million!

Re: Challenging "Find the Hashed Area" Question

Hint: The side length of the larger square is the diagonal of the smaller square.

Re: Challenging "Find the Hashed Area" Question

I understand the rest of it but just can't the length of the inner square. I'm sorry my brain isn't working today. I know it has something to do with the Phytagoreans Theorem. Can you please tell me how you got the side length of the inner square.

Re: Challenging "Find the Hashed Area" Question

A square is composed of two 45-45-90 triangles. The side length of the large square is the diagonal of the small square, so the diagonal of the small square is 6. The diagonal is like the "hypotenuse" of the 45-45-90 triangles, you should be able to find the side length now.

Re: Challenging "Find the Hashed Area" Question

Thank you Richard for the step-wise explanation. I now understand. Since the diagonal of the small square is 6; it would be b^2 + b^2 = 6^2; so solving for b, you would take square root of 36 so 6, then another square root of 6 so "b" (the side of the square) would be 3 sqrt 2. From here, we know that the diameter of the circle is 3 sqrt 2 and we can find radius and then area as you had outlined. Thanks for your help!

Re: Challenging "Find the Hashed Area" Question

Hello, danielmc!

Quote:

A circle is inside a square, which is also found inside another square with with 6 m sides.

Find the shaded area.

Code:

` E`

o

3 * * 3

* *

* *

A * * B

o-------* * *-------o

* | *:::::::::::* | *

3 * | *:::::::::::::::* | * 3

* |*:::::::::::::::::*| *

* |:::::::::::::::::::| *

* *:::::::::::::::::::* *

H o *:::::::::o:::::::::* o F

* *:::::::::::::::::::* *

* |:::::::::::::::::::| *

3 * |*:::::::::::::::::*| * 3

* | *:::::::::::::::* | *

* | *:::::::::::* | *

o-------* * *-------o

D * * C

* *

3 * * 3

* *

o

G

Is the correct answer: $\displaystyle \tfrac{9\pi}{2}\:m^2\,\text{ or }\,9\pi\:m^2\,?$

I'd like to know how you got those two answers . . .

The circle is inscribed in square $\displaystyle ABCD$

. . which is inscribed in square $\displaystyle EFGH$ as shown.

The top triangle $\displaystyle ABE$ is an isosceles right triangle with legs of length 3.

. . Hence: $\displaystyle AB = 3\sqrt{2}$

But $\displaystyle AB$ is the diameter of the circle.

. . Hence, the radius is: .$\displaystyle r \:=\:\frac{3\sqrt{2}}{2}$

Now you can find the area of the circle . . . right?