Puzzle 25 of 26 · Hard

An acute triangle on a circle

You pick three random points on a circle and join them into a triangle. What’s the chance the triangle is acute (every angle under 90°)?

Answering puzzles from Probability as area is part of Pro.

Show the solution

Answer: 1/4 = 25%

The circle opened out into degrees: these three fit in a half circle, so the triangle isn’t acute.

Step 1 · Acute means the centre is inside
$\displaystyle \text{an angle of } 90\text{°} \text{ or more appears exactly when all three points fit in } a \text{ half circle}$
If the points fit in a half circle, the triangle sits to one side of the centre and its middle angle is at least 90°.
Step 2 · Chance all three fit in a half circle
$\displaystyle 3 \div 2^{2} = \frac{3}{4}$
Each point can be the start of a clockwise half circle holding the other two: chance (1/2)² each, and only one point can be it.
Step 3 · Chance the triangle is acute
$\displaystyle 1 - \frac{3}{4} = \frac{1}{4}$
The triangle is acute exactly when the points don’t fit in any half circle.

Answer $\displaystyle \frac{1}{4} = 25\%$

Step 3 of 3

Puzzle 25 of 26 · Hard

You pick three random points on a circle and join them into a triangle. What’s the chance the triangle is acute (every angle under 90°)?

Answering puzzles from Probability as area is part of Pro.

Show the solution

Answer: 1/4 = 25%

The circle opened out into degrees: these three fit in a half circle, so the triangle isn’t acute.

Step 1 · Acute means the centre is inside
$\displaystyle \text{an angle of } 90\text{°} \text{ or more appears exactly when all three points fit in } a \text{ half circle}$
If the points fit in a half circle, the triangle sits to one side of the centre and its middle angle is at least 90°.
Step 2 · Chance all three fit in a half circle
$\displaystyle 3 \div 2^{2} = \frac{3}{4}$
Each point can be the start of a clockwise half circle holding the other two: chance (1/2)² each, and only one point can be it.
Step 3 · Chance the triangle is acute
$\displaystyle 1 - \frac{3}{4} = \frac{1}{4}$
The triangle is acute exactly when the points don’t fit in any half circle.

Answer $\displaystyle \frac{1}{4} = 25\%$

Step 3 of 3