Hello everyone,

First time poster here just wanted to ask you how to solve this question quicker than the way I just did.

Provided info: Given the small circle has radius of 3, what is the area of the shaded region?

The triangle inscribed in smaller circle is split into 3 isosceles triangles with 2 sides equaling 3.

Using the formula x : x : xsqrt2

You get the side of the inscribed triangle is 3sqrt2

Once you have that you can find the area of the equilateral triangle by splitting it into 2 right triangles and using the pythagorean theorem.

Where x = height

((3sqrt2)/2)^2 + x^2 = (3sqrt2)2^2

18/4 + x^2 = 18

9/2 + x^2 = 18

subtract the 9/2

x^2 = 18 - 4.5 = 13.5

x=sqrt13.5

Then you find the area of the triangle by formula (b*h)/2

3sqrt2*sqrt13.5 = 3sqrt27 = 9sqrt3

lastly divide by 2 = 4.5sqrt3

A=4.5sqrt3

Now you know the area of the original smaller triangle.

The area of the smaller circle using A=pi r^2 = 9pi

Now given the ratio of the larger triangle to the smaller triangle 4x

You can find the area of the shaded region of the top part by using the two answers above and divide by 3.

4(4.5sqrt3) - 9pi

(18sqrt3 - 9pi)/3

6sqrt3 -3pi

Since the smaller triangle is 1/4 to the larger triangle, we know the larger circle will be 36pi and we already know the area of the larger triangle = 18sqrt3

So in knowing that we can determine the larger shaded region (at the bottom) by subtracting the area of the larger circle by the larger triangle and then divide it all by 3

(36pi - 18srqt3) / 3

which equals

12pi - 6sqrt3

then we add both shaded areas

6sqrt3 -3pi + 12pi - 6sqrt3

9pi

Can someone tell me if my math is correct and how can you solve this question in under ~2mins?

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