Geometry Problem Solver Polygon shapes, named by their sides
A polygon is a flat, closed shape made only of straight sides. The name comes from the number of sides: three is a triangle, four a quadrilateral, five a pentagon, six a hexagon, and so on. Circles and any shape with a curved or open edge are not polygons.
What makes a shape a polygon
Three conditions, all required. It has to be flat, so a cube is not a polygon although each of its faces is. It has to be closed, so a path that stops without returning to its start is not one. And every side has to be a straight line segment, which is what rules out circles, semicircles and anything with a curve in it.
Regular and irregular
A polygon is regular when all its sides are the same length and all its angles are equal. A square is a regular quadrilateral; a rectangle that is not square is an irregular one. Both are still quadrilaterals, because the name only counts sides.
This trips people up on tests: a shape does not stop being a pentagon because it looks lopsided. Five straight sides, closed and flat, is the whole requirement.
Convex and concave
A convex polygon has every interior angle under 180°, and any line segment drawn between two points inside it stays inside. A concave polygon has at least one interior angle over 180°, which shows up as a corner that caves inward. An arrowhead is the standard example.
Adding up the interior angles
For a polygon with n sides, the interior angles always sum to (n − 2) × 180°. The reasoning is that any polygon can be cut into n − 2 triangles from a single corner, and each triangle contributes 180°.
For a regular polygon, divide that sum by n to get each individual angle. A regular hexagon: (6 − 2) × 180° = 720°, then 720° ÷ 6 = 120° per corner.
Polygons by number of sides
| Sides | Name | Interior angles add to | Each angle if regular |
|---|---|---|---|
| 3 | Triangle | 180° | 60° |
| 4 | Quadrilateral | 360° | 90° |
| 5 | Pentagon | 540° | 108° |
| 6 | Hexagon | 720° | 120° |
| 7 | Heptagon | 900° | 128.57° |
| 8 | Octagon | 1080° | 135° |
| 9 | Nonagon | 1260° | 140° |
| 10 | Decagon | 1440° | 144° |
| 12 | Dodecagon | 1800° | 150° |
Questions
Is a circle a polygon?
No. A polygon is made of straight sides, and a circle has none. A shape with many short straight sides can look like a circle, but that shape is a polygon with a high side count, not a circle.
What is a shape with 4 sides called?
A quadrilateral. Squares, rectangles, rhombuses, parallelograms, trapezoids and kites are all quadrilaterals, distinguished by their angles and side lengths.
Do the exterior angles change with the number of sides?
No, and that is the useful part. The exterior angles of any convex polygon add to 360°, whatever the side count. For a regular polygon, each one is 360° ÷ n.
What is the polygon with the most sides called?
There is no last one. Names exist up to about 20 sides, after which mathematicians normally write n-gon instead: a 17-sided polygon is a 17-gon, and nobody minds.
More geometry reference
Stuck on the actual problem
Photograph it and get the worked solution, step by step, with the theorem named at each one.
