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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