In the diagram, each of the three identical circles touch the other two. The circumference of each circle is 36. What is the area of the shaded region?
https://gyazo.com/0bb2e632441b774f3d412d086fa55957
I made an equilateral triangle using the centers of the triangle.
Since an equilateral triangle has angles of 60o, each sector part is just 16 of the circle.
Circumference of each circle = 36 units
Radius = 18π
Sector areas = 3⋅16⋅(18π)2π
= 12⋅324π2π
= 12⋅324π
= 162π
Now the area of the equilateral triangle is:
√34⋅(36π)2
= √34⋅1296π2
= 1296√34π2
Which means the area of the shaded region is:
1296√34π2−162π
= 1296√34π2−162ππ2
= 1296√34π2−648π4π2
= 1296√3−648π4π2
= 4(324√3−162π)4π2
= 324√3−162ππ2
please tell me if I did anything wrong...
LOL
I got something complicated as well... I thought I got it incorrect. i actually got the same answer as you, so nothing's wrong... I think.
lol I was debating whether or not to post that because there was a similar question, however this was asking for area of the shaded region instead of the perimeter of it...
See image below:
Triangle area - cicle sector area * 3 = area in question
sqrt(3)/4 ( 36/ pi)2 - 3 * 1/6 * pi ( 18/pi)2 = shaded (we use 1/6 because the sector of the circle is 60/360 of the circle)
324 sqrt(3) / pi2 - 162 / pi <============ Same answer as LR Way to go Logarhythm!!