Find the shaded area.
* * *
* |r/2 *
* A| *
* o *
* .:|:. *
* .::O::. *
* .::*|*::. *
* .::*::|::*::. *
* ::*:::|:::*:: *
Bo - - - - - oC
is the center of the circle.
Note that is equilateral.
Its height is: .
The height of is: .
The area of is: .
good job Soroban but I need the blue area
and as you see it is a part of circle not a full
circle to substact the area of triangle !!
Ha! . . . That was stupid!
I found the wrong area . . . *blush*
Let's see, the shaded area is the area of the circle,
. . minus the area of , minus that missing segment.
The area of the segment is the area of the sector
. . minus equilateral triangle
The area of the sector is: .
The area of the equilateral triangle is: .
So the area of the segment is: .
Therefore, the desired area is:
Here is a slightly different method. In the attached picture (where the top of the white equilateral triangle is the centre of the circle), the blue and red regions together form 5/6ths of the area of the circle, so their combined area is . From that we must subtract the red area. Each red triangle has area half the product of two sides times the sine of the included angle. In this case the sides are and , and the included angle is . So the area of each triangle is . Therefore the blue area is , as Soroban already showed.
Sometimes it's worth doing a problem in more than one way, just to check that you have the correct answer.