Polygons

Skills and methods in this topic, easy to hard

1. 1Regular polygonsAll sides equal and all angles equal; n lines of symmetry, order n

   MethodRegular hexagon: 6 lines of symmetry, a fit every 360° ÷ 6 = 60°

   Another way: From the centreAngle at the centre = 360° ÷ n: an octagon's 360° ÷ 8 = 45°

2. 2Angle sum of a polygonAngle sum = (n − 2) × 180°; a regular one's angle = sum ÷ n

   MethodHexagon: (6 − 2) × 180° = 720°; regular, each 720° ÷ 6 = 120°

   Another way: Cut from a point inside6 × 180° − 360° = 720°

3. 3Exterior angles of a polygonExterior angles always make 360°; sides = 360° ÷ one exterior angle

   MethodInterior 140°: exterior 180° − 140° = 40°, n = 360° ÷ 40° = 9

   Another way: An equation from the angle sum(n − 2) × 180 = 140n, n = 9

4. 4TessellationsThe angles at each vertex must make exactly 360°

   MethodHexagons: 3 × 120° = 360°, yes; pentagons: 360° ÷ 108° isn't whole, no

   Another way: Add the angles135° + 135° + 90° = 360°: two octagons and a square

5. 5Equilateral triangles, squares and hexagonsTriangle: two arcs of one radius; square: two diameters at right angles (a set square's corner); hexagon: the radius marked off 6 times round

   MethodAC = BC = AB: three equal sides

   Another way: Six equilateral triangles6 × 60° = 360°, each angle 60° + 60° = 120°

6. 6Drawing a regular pentagon72° at the centre, 108° inside; mark the side CN off five times

   Method∠BAC = (180° − 108°) ÷ 2 = 36°

   Another way: By the exterior angles360° ÷ 5 = 72°, 180° − 72° = 108°

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Regular polygons →

   O is the centre of the regular polygon. x =

   ▶Show the answer and working

   Answer: 30°

   The polygon has 12 sides and 12 equal angles at the centre making a full turn: x = 360° ÷ 12 = 30°.

2. Example 2Tessellations →

   Regular 4-sided polygons alone can tessellate a plane.

   True or false?

   ▶Show the answer and working

   Answer: True

   Each angle of a regular 4-sided polygon is 90°. At every vertex the angles must make exactly 360°, so 360° ÷ the angle must be whole: only triangles, squares and hexagons manage it.

3. Example 3Drawing a regular pentagon →

   O is the centre of the regular pentagon. x =

   ▶Show the answer and working

   Answer: 72°

   The five angles at the centre are equal and make a full turn: x = 360° ÷ 5 = 72°.