Constructions and Tilings

Constructions with ruler and compass, interior angles of regular polygons, and which shapes fit around a point to tile a floor.

Key ideas

Every point on the perpendicular bisector of AB is the same distance from A and B.
An angle bisector splits an angle into two equal angles.
A regular hexagon fits in a circle with each side equal to the radius.
Shapes tile around a point only if their angles there add up to exactly 360°.
The only regular polygons that tile a floor on their own are equilateral triangles, squares and regular hexagons.

Angle sum of an n-sided polygon	(n − 2) × 180°
Each angle of a regular n-gon	(n − 2) × 180° ÷ n
Common angles	triangle 60°, square 90°, pentagon 108°, hexagon 120°, octagon 135°

Using n × 180° for the angle sum.
Thinking regular pentagons can tile a floor (3 × 108° = 324°, which leaves a gap).
Changing the compass width in the middle of a construction.

Can two regular hexagons and two equilateral triangles fit together around one point with no gap or overlap? Show the angle sum.

Working: triangle 60°, square 90°, pentagon 108°, hexagon 120°, octagon 135°, 12-gon 150°

Answer: Yes, 360°