Given an even number N which represents the number of sides of a regular polygon with N vertices, the task is to find the square of the minimum size such that given Polygon can completely embed in the square.
A Polygon is a convex figure and has equal sides and equal angles. All sides have length 1.
Embedding: Place Polygon in the square in such way that each point which lies inside or on a border of N should also lie inside or on a border of the square.
Input: N = 4
Regular polygon with 4 Sides is square with side 1.
Given polygon can easily embed on the square with side 1.
Input: N = 6
Regular polygon with 6 Sides is Hexagon with side 1.
Given polygon can easily embed on the square with side 1.931851653.
Approach: The idea is to observe that on a 3-D plane, when a polygon is embedded in a square, it might be rotated. A similar approach has been discussed in Hexagon problem and Octagon problem . Therefore, we take the projection of each side on both the axis using the mathematical functions sin() and cos(). The overall sum of all the projections is the minimum side of the square required in this problem.
Below is the implementation of the above approach:
Time Complexity: O(N), where N is number of sides of polygon.
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- Length of Diagonal of a n-sided regular polygon
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- Program to find the Interior and Exterior Angle of a Regular Polygon