Polygon angle sums
A polygon is a closed figure made of straight sides. Splitting polygons into triangles gives simple formulas for their angles.
Polygons
A polygon is a closed plane figure formed by three or more segments that meet only at their endpoints. Polygons are named by their number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), octagon (8), decagon (10), and so on. A regular polygon has all sides and all angles congruent.
Convex and concave
A polygon is convex if every interior angle is less than 180 degrees, so no side 'caves in'; any segment joining two interior points stays inside. A concave polygon has at least one interior angle greater than 180 degrees, like an arrowhead shape.
Interior angle sum
Drawing diagonals from one vertex splits an n-sided convex polygon into n - 2 triangles, so its interior angles sum to \((n - 2) \times 180°\). A quadrilateral's angles sum to 360 degrees, a pentagon's to 540, and a hexagon's to 720.
Regular polygon angles
Each interior angle of a regular n-gon is \(\frac{(n - 2) \times 180°}{n}\). A regular hexagon has interior angles of 720/6 = 120 degrees, and a regular octagon, the shape of a stop sign, has angles of 1080/8 = 135 degrees.
Exterior angle sum
Taking one exterior angle at each vertex, the exterior angles of any convex polygon add to 360 degrees, a full turn. Walking around the polygon, you turn by each exterior angle and end facing your starting direction. Each exterior angle of a regular n-gon is 360/n degrees.
Finding the number of sides
If each exterior angle of a regular polygon is 24 degrees, it has 360/24 = 15 sides. If each interior angle is 140 degrees, each exterior angle is 40 degrees, so there are 360/40 = 9 sides. Working with exterior angles is usually simpler.
Diagonals
A diagonal joins two non-adjacent vertices. An n-sided polygon has \(\frac{n(n - 3)}{2}\) diagonals: a pentagon has 5 and a hexagon 9. From each vertex, n - 3 diagonals can be drawn, and dividing by 2 avoids counting each one twice.
Worked example
A floor tiler wants regular tiles that fit together around a point with no gaps. The angles meeting at a point must sum to 360 degrees. Equilateral triangles (60 degrees, six at a point), squares (90, four at a point), and regular hexagons (120, three at a point) work. Regular pentagons, at 108 degrees, do not, since 360/108 is not a whole number, which is why honeycombs are hexagonal rather than pentagonal.
Key takeaways
Polygons are closed figures with straight sides, named by side count; regular polygons have equal sides and angles. Convex polygons have all interior angles under 180 degrees. Interior angles sum to (n - 2) x 180 degrees, so each interior angle of a regular n-gon is that sum divided by n. Exterior angles, one per vertex, always sum to 360 degrees, so each exterior angle of a regular n-gon is 360/n. An n-gon has n(n - 3)/2 diagonals. Only triangles, squares, and hexagons tile a flat surface as regular polygons alone.
Flashcard
Closed figure of straight sides
Check your understanding
What is the sum of the interior angles of a hexagon?