Definition of a Convex Shape
A convex shape, in geometry, is a shape where for any two points chosen inside the shape, the straight line segment connecting these two points lies entirely within or on the boundary of the shape. Informally, a convex shape is one that has no "dents," "holes," or indentations; it is always bulging outwards or perfectly flat along its edges.
Key Properties and Characteristics
A fundamental characteristic of a convex polygon is that all its interior angles are strictly less than 180 degrees. If an imaginary line segment is drawn between any two points on the boundary or within a convex shape, that segment will never leave the confines of the shape. Examples include triangles, squares, circles, and ellipses.
Practical Examples of Convexity
Many everyday objects exhibit convex shapes. A football or a tennis ball is a convex three-dimensional object. In two dimensions, a standard playing card (rectangle), a coin (circle), or a perfectly straight ruler are all examples of convex shapes. When you hold any convex object, no part of its surface curves inward relative to its outer boundary.
Distinguishing Convex from Concave Shapes
The opposite of a convex shape is a concave shape. A concave shape is characterized by having at least one interior angle greater than 180 degrees, or at least one section that curves inward. For a concave shape, it is always possible to find two points within it such that the line segment connecting them passes outside the shape's boundary. A star, a crescent moon, or the shape of a boomerang are common examples of concave shapes.