Interior & Exterior Angles Mathematics Interior & Exterior Angles

Aims of the Lesson To learn key words and facts linked with angles connected with polygons. To find out the general rules for calculating the interior and exterior angles of polygons.

Key Words CONVEX – all corners point outwards CONCAVE – at least one corner point inwards (like it has ‘caved’ in on itself) REGULAR – all sides and angles are equal IRREGULAR – not ALL sides and angles are equal Convex Concave

Angles inside Shapes INTERIOR angles (i) are the angles on the inside of the shape at each of its corners. If all interior angles are equal (and the sides are too) then the shape is regular. i

Finding the Total of the Interior Angles Draw a convex shape. Draw lines from ONLY that corner to every other corner and count how many triangles you have cut it into. Each triangle = 180° Total interior angle = no. of triangles × 180°

Example Using a pentagon... A pentagon (5 sides) splits into 3 triangles 3 × 180° = 540° (total interior) 1 2 3

Interior Angles For regular shapes, to find the value of one interior angle, divide the total interior angle by its number of sides. A regular pentagon (5 equal sides) 540° ÷ 5 = 108° (each interior angle) 108°

Discover the general rules… Draw up a table like the one shown below. Use the instructions given on the previous slides to draw shapes, cut them into triangles and therefore fill-in the first 4 columns of the table. No. of No. of Total Interior One Interior Exterior Sides Triangles Angle Angle Angle 3 4 5 6

Exterior Angles If you extend one side of a polygon, the EXTERIOR angle (e) will be the angle between this extension and the next side… i e

Calculating the Exterior Angle Did you notice that the interior and exterior angle form a straight line? This means they add up to 180°. If you subtract the interior angle from 180 you can calculate the exterior angle. Use this to fill-in the last column of your table.

Alternative Calculations You can also calculate the exterior angle by dividing 360 by the number of sides. You can then calculate the interior angle by subtracting the value of the exterior angle from 180.

Example Can regular octagons tessellate? To answer this question you need to know that tessellate means that they fit together like tiles that do NOT have gaps or overlaps. You need to remember that if they fit together then at the points where they meet their angles add up to 360° exactly. You need to calculate the interior angle of an octagon. An octagon has 8 sides. One exterior angle = 360 ÷ 8 = 45° One interior angle = 180 – 45 = 135° One tile’s angle = 135° 2 tiles = 2 x 135 = 270° 3 tiles = 3 x 135 = 405° Answer: NO  gap of 90°  overlap of 45°

What next? Print out the notes called Angle4. Read through them and make sure you answer any questions. Work through the MyMaths lesson and then its online homework called: Shape > Angles > Interior Exterior Angles found at: http://app.mymaths.co.uk/259-resource/interior-exterior-angles http://app.mymaths.co.uk/259-homework/interior-exterior-angles Shape > Angles > Sum of Angles in a Polygon found at: http://app.mymaths.co.uk/260-resource/sum-of-angles-in-a-polygon http://app.mymaths.co.uk/260-homework/sum-of-angles-in-a-polygon Shape > Angles > Angle Proofs found at: http://app.mymaths.co.uk/261-resource/angle-proofs http://app.mymaths.co.uk/261-homework/angle-proofs Now move on to the Angle5 powerpoint