Polygons

A polygon is a closed flat shape whose sides are all straight line segments. Closed means the boundary joins up with no gaps. Straight rules out anything with a curve, which is why a circle is not a polygon and never becomes one however many points it has.

Names by number of sides

Regular and irregular

A polygon is regular when all its sides are equal and all its angles are equal. Both conditions are needed. A rhombus has four equal sides but unequal angles, so it is not regular. A rectangle has four equal angles but unequal sides, so it is not regular either. A square has both, and is the regular quadrilateral.

Why the angle sum follows a pattern

Any polygon can be cut into triangles by drawing diagonals from one corner. A quadrilateral splits into two triangles, a pentagon into three, a hexagon into four. The number of triangles is always two fewer than the number of sides, and each triangle contributes 180 degrees.

For a regular polygon, dividing that total by the number of sides gives each interior angle. A regular hexagon has 720 / 6 = 120 degrees at every corner, which is why hexagons tile a floor with no gaps.

Common mistakes

Counting an open shape

A shape with a gap in its boundary is not a polygon, however many straight sides it has.

Assuming every hexagon is regular

Textbook pictures are almost always regular, so students come to expect it. A hexagon only needs six straight sides, and it can be lopsided.

Check yourself