okay in this video we're going to look at angles and polygons um what a polygon is polygon is a shape with straight sides i've drawn all the or printed out all the regular polygons here from three sides up until 10 sides so let's just run through those a triangle equilateral triangle is the regular uh polygon uh with three sides regular means all angles are the same size and all the sides of the same side uh quadrilateral that would be a square all the angles the same size as him regular pentagon five sides regular hexagon six sides regular heptagon seven sides regular octagon eight sides regular nanogun nine size uh size sides and a regular decagon 10 sides and the angles add up to in each of these shapes and it adds up to 180 degrees uh quadrilateral 360 pentagon 540 hexagon 720. heptagon 900 an octagon 1000 denier uh nonogon uh 1260 and a decagon 1440 now it's interesting to note that a triangle is 180 degrees you will have seen that um much further down in your in your mouth studies or in previous years a quadrilateral adds up together to give you 360. you should know these two now as you add each side so as you go from four sides to five sides you add on an over 180 as you go from five sides to six sides you add on over 180 six sides to seven sides and over 180 and each time you add a side you add another 180 on so to find out what the angles would add together to give you in a 11 sided chip you would add 180 on to 1440. there is another uh way to work that out that's by using the formula and i'm just going to show you that now okay to find the sum of the interior angles so what the angles inside the shape what they add up to use the formula and take away two ends of the number of sides take away two and then you times about 180 so you take two off the number of sides and then you times it by one earring so for a triangle it's got three sides you take away two is one and then times 180 is 180 so the angles and triangle add up to give 180 for for instance an octagon eight sides eight take away two is six and then you do six times 180 which is one thousand donating if for instance you had a hundred side chip you do a hundred take away to his 98 and times 180 and that will tell you what all the angles inside of the shape would add up to so this formula is quite useful okay so let's have a look at some examples so the question says find the sum of the interior angles of a polygon with 20 sides so remember our formula was n take away 2 times 180 so get your scientific calculator and you're going to take 2 off the number of sides and you're going to times about 180 so you're going to do 20 take away 2 which is 18 press equals that point because remember this has got brackets around it so you you want to do the sum of two separate steps you're going to want to take two off the the to 20 take away two and then times by 180 alternatively could use the brackets on the calculator so 20 take away 2 is 18 and then times by 180 gives an answer off 3240 so the angles in a 20-sided shape add together to give 3240 this sum of the interior angles means what do the angles inside the shape add up together to give okay here's another question it says a regular polygon has 12 sides find the size of each interior angle so in other words it wants to know what each one of the angles we'll add what each one of the angles is so let's find out what they all add up together to give what the sum of the angles are is so n take away 2 times 180 so there's 12 sides so you do 12 take away 2 is 10 times 180 which equals 1 800. that means it degrees that means that all the angles inside of the shape add together to give 1 800. so we need to find the size of each of the interior angles because it's regular it means all the angles are the same so all 12 angles are the same so if you divide 1 800 by 12 you'll find the size of one angle so get your scientific calculator and do 1800 divided by 12. when you do that you get 150 degrees so it means each of the angles is 150 degrees because you do 1 800 divided by 12. sure you're working out there to be 150 degrees okay another question uh the sum of the interior angles of a polygon is 3960 degrees and i want to know how many sides this polygon has so remember the formula n take away 2 times 180 is what the angles inside of the shape are together to give are the sum of the interior angles now we know that is 3 960. so that equals 3960. so we just need to solve this equation so we're going to divide by 180 so we're going to divide both sides by 180 so that would give you n minus 2 equals and then using your scientific calculator you can do 3960 divided by 180 to give 22 so you get 22 then you need to get rid of the minus 2 so we're going to add 2 to both sides so you get n equals 24. if you wanted to use it using um just working backwards remember we took away 2 then times 1 times 180 so if you divide your answer by 180 and then add 2 you'll get your answer just working using the inverses we're not going to look at exterior angles so as you know polygons have interior angles inside the shape they also have exterior angles and if you carry on little the sides and consider the angle bond between that line and the set at the next side they're what we call exterior angles here's a regular pentagon and it's got five exterior angles now inside of the angle so we know that the interior angle is 108 degrees each because if you do 540 divided by 5 you get 108 that means that the outside angle is 72 degrees so first rule is interior angles interior angle plus exterior angle will always add together to give you 180 degrees because the interior angle and the exterior angle make a straight line next thing to consider is here you can see this angle 72 degrees this angle 72 degrees this angle 72 degrees and so on so all five exterior angles are 72 degrees 72 times 5 is 360. that would work for all xl all regular uh all regular polygons if you consider an equilateral triangle the interior angle is the exterior angle oh so the interior angle is 60. the exterior angle would be 120 320 is 360. so all exterior angles sum of exterior angles for regular or irregular polygons is 360 degrees okay here we've got a section of a regular polygon we don't know how many sides it's got because we only can see that it's only that it's got one two three sides you can see that it's marked on an exterior angle here's the interior angle and here's the exterior angle what we need to do is the question has asked us to work out how many sides the shape has got now we know that the sum of the exterior angles or some of the outside angles is 360 degrees so if we add it together this exterior angle the next exterior angle the next one and all the exterior angles they will all add up together to give you 360 degrees so if you divide 360 degrees by the size of each exterior angle which is 20 degrees you should be able to work out how many exterior angles is going to be so if you do 360 divided by 20 you get this 18 exterior angles if there's 18 of these outside angles that means there's 18 angles which means that this the shape has got 18 sides so that means this regular polygon has got 18 sides