Identifying Convex and Concave Polygons
Identifying Convex and Concave Polygons is a Grade 8 math skill from Big Ideas Math, Course 3, Chapter 3: Angles and Triangles. A polygon is convex if all interior angles are less than 180 degrees and every segment connecting two vertices lies inside the polygon, while a concave polygon has at least one interior angle greater than 180 degrees and some connecting segments pass outside the boundary. Recognizing this distinction is important before applying interior angle sum formulas.
Key Concepts
A polygon is convex if for every pair of vertices, the line segment connecting them is entirely inside or on the polygon. All interior angles of a convex polygon measure less than $180^\circ$.
A polygon is concave if there is at least one pair of vertices for which the line segment connecting them goes outside the polygon. A concave polygon has at least one interior angle greater than $180^\circ$.
Common Questions
What is the difference between a convex and concave polygon?
A convex polygon has all interior angles less than 180 degrees and all diagonals inside; a concave polygon has at least one interior angle greater than 180 degrees and some diagonals outside the shape.
How can you tell if a polygon is concave?
If any interior angle is a reflex angle (greater than 180 degrees), or if you can draw a line segment between two vertices that passes outside the polygon, it is concave.
Why does it matter if a polygon is convex or concave?
Interior angle sum formulas and many geometric properties apply specifically to convex polygons, so identifying the type is the first step in solving problems.
Where are convex and concave polygons covered in Grade 8 Big Ideas Math?
Big Ideas Math, Course 3, Chapter 3: Angles and Triangles includes classifying polygons as convex or concave.