there are four circles. 1 is a big circle and inside of that circle are the 3 remaining circles(the 3 have the same size). each of the four circles is tangent to the other three. if the radius of the smaller circles is a, find the area of the largest circle.
please show solutions clearly.. i am really confused.
please show solutions clearly.. i am really confused.
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Okay it's a little bit hard to explain without a diagram but here it goes:
Let a = radius of smaller circle, R = radius of bigger circle
Connect the centers of the smaller circles and you'll get an equilateral triangle with each side 2a.
If you draw medians of this triangle, the intersection will be the center of the bigger circle.
You'll see that the equilateral triangle is split into three isosceles triangle (30°, 30°, 120°) with the longer side being 2a. Here's my artistic rendering lol: http://tinyurl.com/7l2q8qm
Notice that R = a + x
From the diagram you see that x = a / cos30
x = 2a / (sqrt3)
Therefore, R = a + (2a/sqrt3) = a(2+sqrt3)/(sqrt3)
Let a = radius of smaller circle, R = radius of bigger circle
Connect the centers of the smaller circles and you'll get an equilateral triangle with each side 2a.
If you draw medians of this triangle, the intersection will be the center of the bigger circle.
You'll see that the equilateral triangle is split into three isosceles triangle (30°, 30°, 120°) with the longer side being 2a. Here's my artistic rendering lol: http://tinyurl.com/7l2q8qm
Notice that R = a + x
From the diagram you see that x = a / cos30
x = 2a / (sqrt3)
Therefore, R = a + (2a/sqrt3) = a(2+sqrt3)/(sqrt3)
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A,B,C is center of 3 identical circle --> triangle ABC is equilateral , O is the centroid of ABC and it also is the center of the larger circle , now it's easy to find the radius