Expected Area of a Triangle

Calculus Level pending

It is widely known that a triangle inscribed in a semicircle with the base of the triangle being the diameter of the semicircle is always a right triangle.

A A and B B are the endpoints of the diameter of a circle with radius 1. 1. Point C C is a point chosen with equal probability along the circumference of the circle. The expected area of Δ A B C \Delta ABC is N . N. What is ⌊ 100 N ⌋ ? \lfloor 100N \rfloor\text{?}


The answer is 63.

Trevor B. by Trevor B. , 22, USA

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1 solution

Kenny Lau
Jul 11, 2014
  • Let the center of the circle be O O .
  • Let ∠ C O B \angle COB be θ \theta .
  • The area of the right triangle formed is ∣ sin ⁡ θ ∣ |\sin\theta| .

N = 1 2 π ∫ 0 2 π ∣ sin ⁡ θ ∣ d θ = 1 2 π ∫ 0 π sin ⁡ θ d θ − 1 2 π ∫ π 2 π sin ⁡ θ d θ = 1 2 π { [ − cos ⁡ θ ] 0 π − [ − cos ⁡ θ ] π 2 π } = 1 2 π { 1 − ( − 1 ) − ( − 1 ) + 1 } = 4 2 π \begin{array}{rcl} N&=&\frac1{2\pi}\int_0^{2\pi}|\sin\theta|\mbox{ d}\theta\\ &=&\frac1{2\pi}\int_0^\pi\sin\theta\mbox{ d}\theta-\frac1{2\pi}\int_\pi^{2\pi}\sin\theta\mbox{ d}\theta\\ &=&\frac1{2\pi}\left\{[-\cos\theta]_0^\pi-[-\cos\theta]_\pi^{2\pi}\right\}\\ &=&\frac1{2\pi}\{1-(-1)-(-1)+1\}\\ &=&\frac4{2\pi} \end{array}

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