Combination of concepts

Geometry Level 3

$\large a - \dfrac{\pi^b}{c\sqrt c}$

The probability of a point within an equilateral triangle with side 1 unit lying outside its in-circle can be written as the above expression for positive integers $a, b$ and $c$ . Find $a+b+c$ .

The answer is 5.

1 solution

Roger Erisman
Nov 28, 2016

Area of equilateral triangle with side = 1 is $\sqrt{3}/4$ . ##### [ A(triangle) = 1/2 b h = 1/2 * 1 * sqrt(3)/4 ]

Area of incircle is pi / 12. ##### [ A(circle) = pi* r^2 = pi * (1 / (2*sqrt(3))^2 ]

Area outside of incircle is sqrt(3) / 4 - pi / 12 = ( 3*sqrt(3) - pi ) / 12

Probability = Area outside of incircle / area of triangle =[ (3 sqrt(3) - pi ) / 12 ] . (3 sqrt(3) / 4 )

This gives P = (3 * sqrt(3) - pi ) / (3*sqrt(3))

Simplifying gives P = 1 - pi/ (3*sqrt(3) so a = 1, b = 1, c = 3 which gives a + b + c = 5.

Very simply presented :)

Calvin Lin Staff - 4 years, 6 months ago

Combination of concepts

The answer is 5.

1 solution

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