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Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.

 

An equilateral triangle, a regular dodecagon, a square, and a regular $n$-gon, all with the same side length, also completely surround a point. Find $n$.

 Dec 24, 2023
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Each interior angle of a regular pentagon measures 180(5−2)/5=108∘.

 

Each interior angle of a regular decagon measures 180(10−2)/10=144∘.

 

Since the two pentagons and the decagon completely surround a point, the sum of their angles must be 360∘.

 

Let x be the side length of the pentagons and decagon. We can form an equation: 2⋅108∘+144∘=360∘. Solving for x gives x=3515​​​.

 

Equilateral triangle, dodecagon, square, and -gon:

 

Each interior angle of an equilateral triangle measures 60∘.

 

Each interior angle of a regular dodecagon measures 180(12−2)/12=150∘.

 

Each interior angle of a square measures 90∘.

 

Since all four shapes completely surround a point, the sum of their angles must be 360∘.

 

Let y be the side length of the triangle, dodecagon, square, and -gon. We can form an equation: 60∘+150∘+90∘+(180−2)(x)=360∘. Solving for x gives x=4y600−75y​.

 

Since both angles sum to 360∘ and use the same side length, 3515​​=4y600−75y​. Solving for y gives y=5815​​​.

 

Therefore, the -gon has (180−2)⋅5815​​=216∘​ as its interior angle measure.

 Dec 24, 2023

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