Polygons A polygon is a 2-D shape made when line segments enclose a region. A The corners are called vertices. One is a vertex. B The line segments are called sides. E C D 2-D stands for two-dimensional. These two dimensions are length and width. A polygon has no thickness.

Polygons Teacher notes Define these key terms. Define a regular polygon as having equal sides and equal angles. Explain that in a concave polygon some of the interior angles are reflex angles. The regular polygon is also convex.

Polygons Teacher notes Define these key terms, using a condition and a shape for each category. Define a regular polygon as having equal sides and equal angles. Explain that in a concave polygon some of the interior angles are reflex angles. The regular polygon is also convex.

Naming polygons Teacher notes Pupils should learn any of the names that they do not know already.

Polygons

Tiling patterns Different cultures have tiled their walls in beautiful ways. These tiles are from the Alhambra Palace in Spain. Teacher notes Pupils should be able to find: two types of 16 sided stars (hexadecagon - look up on the internet), 3 types of hexagon, octogons, nonagons, 10 - sided stars (decagons), dodecagons and tridecagons. Pupils do not need to know all these terms but should be able to describe shapes in terms of their number and lengths of sides. Photo credit: © Jennifer Stone 2010, Shutterstock.com Describe the polygons you see.

Interior angles in a triangle

Interior angles in quadrilaterals

Interior angles in polygons Teacher notes Emphasize that n is the number of sides/vertices of the polygon. Ask pupils to practice splitting various polygons into triangles. Ask why the rule always works.

Interior angles in regular polygons A regular polygon has equal sides and equal angles. The size of the interior angles in a regular polygon can be calculated as follows: Name of regular polygon Sum of the interior angles Size of each interior angle Equilateral triangle 180° 180° ÷ 3 = 60° Square 2 × 180° = 360° 360° ÷ 4 = 90° Teacher notes Ask pupils to complete the table for regular polygons with up to 10 sides. Regular pentagon 3 × 180° = 540° 540° ÷ 5 = 108° Regular hexagon 4 × 180° = 720° 720° ÷ 6 = 120°

Interior angles in regular polygons

Polygon exterior angles Teacher notes Emphasize that n is the number of vertices of the polygon which is equal to the number of sides.

Exterior angles An architect has used regular pentagons in her designs for two new buildings. x x Teacher notes 360 ÷ 5 = 72° or 180 – 108° = 72 (108° is the size of the interior angle calculated by n – 2 × 180°) 2 (360 ÷ 5) = 144° or 360 – (2 × 108) = 144° (108° is the size of the interior angle calculated by n – 2 × 180°) Photo credits: Grass: © apdesign 2010, Shutterstock.com Building texture: © MTTrebbin 2010, Shutterstock.com Calculate angle x in each of the aerial views.

Find the number of sides Teacher notes Challenge pupils to find the number of sides in a regular polygon given the size of one of its interior or exterior angles. Establish that if given the size of the exterior angle, divide this number into 360° to find the number of sides. This is because the sum of the exterior angles in a polygon is always 360° (see slide 16) and each exterior angle is equal. Establish that if given the size of the interior angle, divide 360° by (180° – the size of the interior angle) to find the number of sides. This is because the interior angles in a regular polygon can be found by subtracting 360° divided by the number of sides from 180°. The formula (n – 2) 180° = n (interior angle), can be rearranged to give: 180°n – (interior angle)n = 360°. Therefore, 360° /(180° – interior angle) = n

Tiling patterns Create your own tiling pattern using regular polygons. Teacher notes Examples of polygons which fit together are: Interior angles of shapes Triangle: 60° Square: 90° Pentagon: 108° Hexagon: 120° Octagon: 135° 6 × equilateral triangle 4 × square 2 × square and hexagon and equilateral triangle 2 × octagon and square 2 × pentagon and a decagon Pupils will need to experiment with the internal angles, explaining how they know they are correct. Which shapes fit together at a point with no gaps? You can use more than one of the same shape or combinations of shapes at any size.

Regular polygons? Which of these exterior angles are from regular polygons? 10 15 25 30 35 45 50 60 70 72 75 80 90 For the angles which are from regular polygons, what are the number of sides? Find other exterior angles from regular polygons. Teacher notes The regular polygons are: 10 (36 sides) 15 (24 sides) 30 (12 sides) 45 (8 sides) 60 (6 sides) 72 (5 sides) 90 ( 4 sides) Other examples are: 1 (360 sides) 5 (72 sides) 9 (40 sides) 18 (20 sides) etc Pupils need to look for factor pairs of 360: 2 x 180 = 360 leads to a 180 sided polygon with exterior angles of 2 3 x 120 = 360 leads to a 120 sided polygon with exterior angles of 3 Is there a way of finding every whole number exterior angle?