A star shape is generally considered a concave polygon. Understanding this classification requires grasping geometric definitions of convexity and concavity. These concepts categorize shapes based on their internal structure and how line segments connecting points within them behave.
Understanding Convex Shapes
A shape is defined as convex if, for any two points chosen within its boundaries, the straight line segment connecting them remains entirely inside the shape. This means no part of the line segment extends beyond the shape’s perimeter. Imagine stretching a rubber band around a convex shape; it would perfectly follow the outline without any inward bends. Common examples include a circle, square, or triangle, where all interior angles are less than 180 degrees. In a convex polygon, all vertices point outwards, away from the interior.
Understanding Concave Shapes
In contrast, a shape is classified as concave if it possesses at least one indentation or inward curve. Formally, a shape is concave if it is possible to find two points within its boundaries such that the straight line segment connecting them passes outside the shape at some point. This characteristic also means a concave polygon will have at least one interior angle greater than 180 degrees. Examples include a crescent moon, a boomerang, or the letter ‘C’. A triangle, however, can never be concave because its three interior angles always sum to exactly 180 degrees, meaning none can exceed this value.
The Geometry of a Star
A typical five-pointed star is a concave polygon. Its distinctive “inward angles” located between each of its points prevent it from meeting the criteria of convexity. One can easily demonstrate a star’s concavity by drawing a line segment between two points within its form, such as from the tip of one point to the inward angle opposite it. This line segment would inevitably pass outside the star’s boundary before re-entering it.
The interior angles at the “valleys” or re-entrant corners of a star shape are greater than 180 degrees. This property is sufficient to classify any polygon as concave. While the individual points of a star might appear convex, the overall structure, with its characteristic inward bends, firmly establishes it as a concave figure.