How to Determine if a Polygon is Concave: A Step-by-Step Guide

concave polygon

A concave polygon is a polygon that has at least one interior angle greater than 180 degrees

A concave polygon is a polygon that has at least one interior angle greater than 180 degrees. In other words, it is a polygon that “caves in” or has a portion that is not convex.

To determine if a polygon is concave, you can follow these steps:

1. Identify the vertices of the polygon and label them as A, B, C, and so on, in a clockwise or counterclockwise direction.
2. Select any three consecutive vertices of the polygon, for example, A, B, and C.
3. Calculate the angle formed at vertex B using the formula:
angle B = atan2(yC – yB, xC – xB) – atan2(yA – yB, xA – xB)
Here, atan2 is the four-quadrant inverse tangent function, and (xB, yB), (xA, yA), and (xC, yC) are the coordinates of vertices B, A, and C respectively.
4. Repeat step 3 for every three consecutive vertices, i.e., for B, C, and D; C, D, and E; and so on, until you finish with the last set of three consecutive vertices.
5. If any calculated angle is greater than 180 degrees, the polygon is concave. If all angles are less than or equal to 180 degrees, the polygon is convex.

For example, let’s say we have a polygon with vertices A(0, 0), B(1, 1), C(2, 0), D(1, -1), and E(-1, -1).

Using the angle formula, we calculate:

angle B = atan2(0 – 1, 2 – 1) – atan2(0 – 1, 0 – 1) ≈ 0.79 radians ≈ 45.17 degrees
angle C = atan2(-1 – 0, 0 – 2) – atan2(0 – 0, 1 – 2) ≈ 3.93 radians ≈ 225.34 degrees
angle D = atan2(-1 – (-1), 1 – 0) – atan2(0 – (-1), (-1) – 0) ≈ 4.71 radians ≈ 269.99 degrees
angle E = atan2(-1 – (-1), -1 – 1) – atan2(-1 – (-1), (-1) – 1) ≈ 3.93 radians ≈ 225.34 degrees

Since angles C, D, and E are greater than 180 degrees, this polygon is concave.

Remember, a concave polygon can have both concave and convex sides, but it has at least one angle greater than 180 degrees.

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