Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.
An equilateral triangle, a square, and a regular $n$-gon, all with the same side length, also completely surround a point. Find $n$.
Here's how to find the value of n for the equilateral triangle, square, and regular n-gon:
Analyze the pentagons and decagon:
Two pentagons and a decagon can surround a point because the angles at each vertex of the decagon can be paired with two non-consecutive angles of a pentagon, forming a straight line through the central point.
Apply similar logic to the triangle and square:
Consider the equilateral triangle. For this shape to surround a point with the square and n-gon, each vertex of the triangle must coincide with a corner of the square and a vertex of the n-gon.
Similarly, for the square, each vertex must coincide with a corner of the equilateral triangle and a vertex of the n-gon.
Since each vertex of the triangle coincides with a corner of the square, the three angles at that vertex (all 60° in the triangle) must add up to 360°, the angle at the corresponding vertex of the n-gon.
Therefore, the internal angle at each vertex of the n-gon is 360° / 3 = 120°.
A regular polygon with an internal angle of 120° is an equilateral triangle or a hexagon. However, an equilateral triangle already fills one of the spaces, so the n-gon must be a regular hexagon.
Therefore, the value of n is 6. A regular hexagon, along with an equilateral triangle and a square, with all sides the same length, can perfectly surround a point, similar to the two pentagons and a decagon.
This solution demonstrates how understanding the geometric relationships between different shapes and their angles can help solve spatial reasoning problems.