Convex polygon

Definition

A convex polygon is a polygon in which all interior angles are less than 180 degrees and any line segment drawn between two points within the polygon lies entirely inside it. This characteristic ensures that no vertices point inwards, making convex polygons a fundamental shape in geometry, particularly in the study of polygons, perimeter, and tessellations.

5 Must Know Facts For Your Next Test

1. In a convex polygon, the line segment connecting any two points within the shape will never leave the shape, ensuring that it maintains a 'bulging out' appearance.
2. The sum of the interior angles of a convex polygon can be calculated using the formula $(n-2) \times 180^\circ$, where n is the number of sides.
3. All regular polygons are convex, but not all convex polygons are regular; some can have sides of varying lengths.
4. Convex polygons can be easily used in tessellations since their shapes can fit together without leaving gaps, making them ideal for tiling surfaces.
5. Examples of convex polygons include triangles, quadrilaterals, pentagons, and hexagons, each defined by their number of sides.

Review Questions

• How does the definition of a convex polygon distinguish it from other types of polygons?
  • A convex polygon is defined by its property that all interior angles are less than 180 degrees and any line drawn between two points inside it remains within the shape. This distinguishes it from concave polygons, which have at least one angle greater than 180 degrees and inward-pointing vertices. Understanding these differences helps in classifying polygons and applying appropriate geometric principles to solve problems.
• Discuss how the properties of convex polygons contribute to their ability to tessellate surfaces effectively.
  • Convex polygons have sides and angles that allow them to fit together neatly without overlaps or gaps when tessellating. This is due to their outward-bulging nature, which ensures that when placed next to one another, they create a continuous surface. Regular convex polygons like squares and equilateral triangles are particularly effective in creating repetitive patterns in tessellations.
• Evaluate the implications of using convex polygons in geometric design compared to using concave polygons.
  • Using convex polygons in geometric design offers significant advantages such as ensuring stability and simplicity in calculations related to area and perimeter. Unlike concave polygons, which can create complex and unpredictable shapes due to inward angles, convex polygons maintain consistent properties that facilitate easier application of geometric principles. This reliability makes them ideal for various practical applications in architecture and art where structural integrity is crucial.

Related terms

Concave polygon: A concave polygon is a polygon that has at least one interior angle greater than 180 degrees, resulting in at least one vertex that points inward.

Regular polygon: A regular polygon is a convex polygon with all sides and angles equal, such as an equilateral triangle or a square.

Tessellation: Tessellation refers to the covering of a plane with one or more geometric shapes, called tiles, without overlaps or gaps, often involving convex polygons.