In this Grade 8 lesson from Big Ideas Math, Course 3 (Chapter 3: Angles and Triangles), students learn how to find the sum of interior angle measures of polygons using the formula S = (n − 2) · 180°, and discover that the sum of exterior angle measures of any convex polygon is always 360°. Students also practice distinguishing between convex and concave polygons and apply these concepts to find missing angle measures in figures such as pentagons, hexagons, and heptagons.
Section 1
Identifying Convex and Concave Polygons
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 .
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 .
Section 2
Polygon Interior Angle Sum Formula
The sum of interior angles of any polygon with sides is given by:
This formula is derived by dividing any polygon into triangles from one vertex.
Section 3
Polygon Exterior Angle Sum Property
An exterior angle of a polygon is formed by extending one side of the polygon at a vertex. The sum of all exterior angles of any convex polygon is always .
Expand to review the lesson summary and core properties.
Section 1
Identifying Convex and Concave Polygons
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 .
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 .
Section 2
Polygon Interior Angle Sum Formula
The sum of interior angles of any polygon with sides is given by:
This formula is derived by dividing any polygon into triangles from one vertex.
Section 3
Polygon Exterior Angle Sum Property
An exterior angle of a polygon is formed by extending one side of the polygon at a vertex. The sum of all exterior angles of any convex polygon is always .