For regular polygons inscribed in a circle of radius 1, use S(6)=1 to conclude that: S(12)=sqrt{2-sqrt{3}}, S(24)=sqrt{2-sqrt{2+sqrt{3}}}.

trumansoftjf0

trumansoftjf0

Answered question

2022-11-11

Computing the sides of inscribed polygons
For regular polygons inscribed in a circle of radius 1, use S ( 6 ) = 1 to conclude that:
S ( 12 ) = 2 3
S ( 24 ) = 2 2 + 3

Answer & Explanation

mentest91k99

mentest91k99

Beginner2022-11-12Added 17 answers

Step 1
For a hexagon, each of the six triangles that define the inscribed polygon is an equilateral triangle. Thus, the radius of the circle is 1, and the angle subtended is π / 3. To double the number of sides, we half the angle, and use
S ( 12 ) = 2 sin π 12 = 2 1 cos ( π / 6 ) 2 = 2 3
Step 2
For 24 sides, half again:
S ( 12 ) = 2 sin π 24 = 2 1 cos ( π / 12 ) 2 = 2 2 + 3

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