Suppose a circle is inscribed in an equilateral triangle with side length two units.
Another circle is inscribed in the upper corner.
It touches two sides of the triangle and the circle.
Find the area A between the smaller circle and the upper corner of the triangle.
For the Area, I got: √3/27 - pi/81 which is about 0.0253649
Label the top vertex A, with B and C being the other 2; BC is horizontal.
The large and small circles, centered at O and P, have radii R and r; and they intersect at point I. A vertical line from A through O meets BC at point J.
From O, draw a perpendicular line to AC where it intersects at D.
Drawing from O to C shows that DC = R*√3
So AD is 2 - R*√3
But drawing AOD shows that AD = R*√3
AD + DC = R*√3 + R*√3 = 2, so R = √3/3
Draw a horizontal tangent to the top of the small circle making a small equilateral triangle AB'C' with the small circle inscribed.
A similar analysis shows small r = √3/9 and AB'C' has side length 2/3.
Area of AB'C' is s^2*√3/4 = (4/9)(√3/4) = √3/9
Area of small circle is pi*(√3/9)^2 = pi/27
One third of the difference, since there are 3 "corners":
Requested Area = √3/27 - pi/81
About 0.0253649
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Posted by Larry
on 2023-07-25 11:58:10 |
I got ...
