Angles in Polygons GCSE Maths

polygon is closed shape with straight sides for example these shapes here are all polygons you can see they're made with straight sides and the shapes are closed these two shapes here are not polygons because they're not closed you can see we have some gaps here these two shapes are not polygons either because some of their sides are curved we can split polygons into two categories we have polygons that we call regular polygons and also irregular polygons these polygons are regular what makes them regular is that all of their sides are equal in length and additionally all of the interior angles are equal in size if either of these statements are not true then the polygon would be irregular for example this rectangle here all of the interior angles are equal but the sides are clearly not equal in length this rhombus here all of the sides are equal in length but the interior angles are not equal and for these two polygons here the interior angles are not equal and nor are the sides equal in length if we have regular polygon with three sides we call this an equilateral triangle if we extend it to regular polygon with four sides then this is square if we add fifth side then this one would be regular pentagon if we had sixth side this one would be regular hexagon seventh side means we end up with regular heptagon an eighth side now we have regular octagon if we had ninth side we have regular nonagon and if we go up to temp side we have regular deagan for your exams you need to recognize and learn the names of all of these apart from seven and nine-sided shapee so you don't need to know the regular heptagon or the regular nonan but you do you need to know all of the other ones this doesn't mean they won't come up in questions but if they do they'll probably be named for you you may already know that the angles inside triangle add up to make 180° you may also know that the angles inside four-sided shape quadrilateral add up to 360 but many people don't know why the reason it's 360 is we can split quadrilateral into two triangles like this each of the angles in these triangles will add up to 180° so we have two of 180° which is 360 we can extend this idea to shapes with any number of sides so if we increase the number of sides again and look at the angles inside pentagon we can split the Pentagon into three triangles like this each of those triangles will have 180° in them so this time we have three lots of 180° 3 * 180 is 540 so this means the angles inside pentagon add up to 540° let's do another side and go for hexagon this time we can split the shape into four triangles each of those will have 180° in them and then if we add up all of those 180° we get 720 so the angles inside hexagon add up to 720° so the first shape the triangle had three sides and it might seem little odd to say this but it was one triangle the second shape was quadrilateral which had four sides and we split that one into two triangles the Pentagon had five sides and we split that into three triangles the hexagon had six sides and we split that into four triangles notice how the number of triangles is always two less than the number of sides so for the triangle it had three sides and if you subtract two from this you get one triangle the quadrilateral had four sides and if you subtract two from that you get two triangles the Pentagon had five sides if you subtract two you get three triangles and the hexagon had six sides and if you subtract two you get four triangles so the number of triangles we can split shape into is always too less than the number of sides it has we can extend this now to look at shapes with any number of sides for example this shape here now if we use our original strategy and split this into triangles by drawing them this one gets quite messy there are loads of triangles instead we can just count up the number of sides This shape has 15 sides if we subtract two from this we know without drawing them they must be 13 triangles each of those triangles will have 180° in them so we do 13 * 180 which gives you 2340 so all of the angles in this 15-sided shape must add up to 2340 we have special name for this total it's known as the interior angle sum because it's the sum of all of the angles on the inside of the shape we can create formula for this based on what we've looked at so far let's say that we have shape with sides to find the number of triangles we subtract two from the number of sides so this is Sub two and then once we have this number of triangles we just multiplied it by 180 so this formula here will allow you to work out the interior angle sum of any polygon of any number of sides let's have look at how we can do this in some questions so for the first question we're going to work out the interior angle sum of decagon earlier we said decagon had 10 sides so using this formula we want to do 10 subtract two so 10 becomes the and then multiply this by 180 if you do this with calculator you end up with 1,440 or we could work out the interior angle sum of polygon with 24 sides so all we do is replace the in the formula with 24 so it's 24 subtract 2 multipli by 180 and this gives you 3,960 sometimes exam questions may require us to work out the interior angle sum for example this question here for this question we're looking to find the angle we can see that this shape is Pentagon since it has five sides we can work out the interior angle sum of the Pentagon using the formula let's replace the in the formula with five so it's 5 subtract 2 multiplied 180 which gives you 540 this means all of the angles on the inside of that Pentagon must add up to make 540° now can see three of them and we also have one in the top right which is right angle and right angle is 90° if we add up the four angles that we have 95 121 90 and4 4 we get total of 410 but we know that all five of the angles must add up to make 540 so if we subtract this 410 from 540 we'll find the remaining angle 540 subtract 410 is 130 so the size of angle must be 130° now let's try another example like this but slightly more complicated so we've got this irregular polygon here and we're told that angle ABC is 3 * angle BCD and we've has to find the size of angle ABC so angle ABC is this one and angle BCD is this one let's call angle BCD now since angle ABC is three lots of angle BCD it's three lots of and three lots of is 3x next we want to work out the interior angle sum This shape has six sides so we need to do six subtract 2 multiplied by 180 which will give you 720 this means all six of these angles must add up to give 7 12° so let's add up all of the angles now this time two of the angles are algebraic but that's okay we can still add those up so let's start with those two 3x + and then we'll add in all of the numerical angles we have all of this must add up to give you 720 the interior angle sum this 3x + 1 will give you 4X if you add up all of these numbers here you get 496 so we've got 4x + 496 must give us this is now just an equation to solve if we subtract 496 from both sides on the left we'll get 4X and if we subtract it from the right we end up with 224 then we can divide both sides by four on the left this gives you and on the right 224 / 4 is 56 so we find that is equal to 56 but this isn't the answer to the question the question wanted us to find the size of angle ABC which is this one here and we labeled that one as 3x so if we know that is 56° 3x must be three lots of this so 3 * 56 gives you 168° and that is the answer to the question next we're going to take closer look at some regular polygons like this regular pentagon here if we calculate its interior angle sum by doing 5 subtract 2 * 180 we get 540 this means all of the angles inside this shape must add up to make 540 but since it's regular remember that all of these angles must be the same size so we can actually find out the size of each of these angles by dividing the sum by how many angles there are the sum is 540 and there are five angles so 540 ID 5 gives you 108 this means that all of these angles must be exactly 108° we can do this for any regular polygon for example this regular hexagon here this time we do the interior angle sum 6 subtract 2 * 180 which gives you so all of the angles must add up to make 720 but there are six of them and they're equal in size so we can find one by doing 720 / 6 which gives you 120° so each of these angles is 120 since these angles are on the inside of the shape we call each of them the interior angle so we can now create formula for the interior angle of regular polygon we started by finding the interior angle sum and then we divided this by how many angles there were which is also the same as the number of sides on the shape which we use the letter for so we just divide the interior angle sum by let's see how we can use this formula in some questions so for this question we're given regular polygon and here it is and we're asked to work out the size of angle marked so since this is regular polygon we can use the interior angle of regular polygon formula at the top we need to know how many sides This shape has and if you count them there are 12 so we're just going to replace the in the formula with 12 so we have 12 subtract 2 * 180 which would normally give us the interior angle sum but since we're trying to find Just One of the angles we divide this by how many angles there are which is 12 if you type this into the calculator you'll get 150° so the answer for is 150° what about this question here work out the interior angle of regular decagon we said earlier that decagon had 10 sides so we're just going to replace the in the formula with 10 so we do 10 subtract 2 * 180 which would normally give us the interior angle sum but we want just one of the angles so we divide it by 10 since there are 10 of them if you type this into your calculator you'll get 144° and let's try this question here work out the interior angle of regular polygon with 30 sides so all we do here is replace the in the formula with 30 30 subtract 2 * 180 IDE 30 typing this into your calculator will give you the answer of 168° so this is how you find the interior angle of regular polygon and it only works if the polygon is regular next we're going to take look at exterior angles if we bring back this regular pentagon this angle here was known as the interior angle because it was on the inside so most people tend to think that this would be an exterior angle because it's on the outside but unfortunately that's not how we draw an exterior angle to create an exterior angle we extend the length of one of the sides it's then the angle that this extended line makes with the next side along so this angle Here is known as the exterior angle and you could do this with any of the sides of the shape so all of these are also exterior angles so how would we go about finding the size of one of these angles well if we take all of these exterior angles and bring them together like this you'll see it makes circle in full turn there are 360° since there were five angles and there equal in size we could find the size of one of them by doing 360 / 5 this will give you 72° so each of these exterior angles here for regular pentagon is 72° let's have look at how we can apply this to other shapes so what if it was regular hexagon with six sides well here are all of the exterior angles and if we bring all of those together they once again make full turn of 360° well for this one there are six equal angles so we divide 360 by 6 which gives you 60° so each of these angles here is 60° notice how it doesn't matter how many sides the shape has the exterior angles always add up to 360 this means we can easily write formula for calculating the size of the exterior angle so the exterior angle for regular polygon is just 360 divide by how many sides it has which we called let's have look at how we can use this formula to solve some problem problems so first of all we've got work out the exterior angle of regular octagon earlier we said that an octagon had eight sides so we just need to replace the in the formula with 8 it's as simple as 360 / 8 which gives you 45° and what about if we were asked to work out the exterior angle of regular polygon with 15 sides well we just do 360 IDE 15 which gives you 24° so working out the exterior angle of regular polygon is very straightforward we just divide 360 by however many sides it has let's bring back the regular pentagon from before we worked at its interior angle at 108° and we also found the exterior angle which was 72° if you add together 108 and 72 you get 180° so for this shape the interior angle plus the exterior angle makes 180° and this is not surprising since they're both angles on straight line in fact this formula interior angle plus exterior angle equal 180 works for any regular polygon we can use this in some questions to help speed up the process for example if we are asked to work out both the interior and exterior angles of regular polygon with 18 sides well the exterior is easy to work out we just do 360 / by where is the number of sides so in this case 18 so for the exterior angle 360 / by 18 gives you 20° so the exterior angle is very quick to find we learned formula earlier for finding the interior angle it was his formula here so to find the interior angle we replace with 18 so 18 subtract 2 * 180 divide by 18 which if you type into your calculator will give you 160 but we can use this formula here at the top to speed up this process if you already know the exterior angle we can quickly find the interior angle because they must both add up to 180 so for this shape the exterior angle was 20° so we can just subtract this from 180 which gives us 160 and that's lot easier and quicker to do now there's one final formula that we need to cover for angles and polygons if we take the formula for the exterior angle exterior angle was 360 divid by the number of sides if we rearrange this formula then the number of sides must be 360 divid by the exterior angle this formula can therefore be helpful in finding the number of signs polygon has if we know its exterior angle let's have look at how we use this in some questions so the exterior angle of regular polygon is 10° work out how many sides it has well to work out the number of sides we just leave 360 / by the exterior so 360 IDE by the exterior which in this question is 10 360 ID 10 is 36 so this shape must have 36 sides let's have look at slightly harder one the interior angle of regular poly is 171 and we need to work out how many sides this one has well the formula requires us to do 360 / the exterior angle but in this question we were told the interior angle but we do know that the interior plus exterior makes 180 so we can find the exterior by subtracting the interior from 180 so if we do 180 subtract 171 the interior we find the exterior is 9° we can now use the formula at the top the number of sides is 360 / by the exterior so 360 / by the exterior which is 9 gives you 40 sides so far in this video we've had five different formulas we've had the interior angle sum the interior angle of regular polygon the exterior angle of regular polygon the interior plus exterior is 180 and also the formula to find the number of sides we're now going to look at how exam questions use some of these formul so here we have diagram and we're told it shows two regular pentagons and regular polygon we're asked to work out the number of sides that regular polygon has so we're going to start with the fact that there are two regular pentagons so these have five sides we're going to work out their interior angles which I've marked onto the diagram here in red we'll do that using the formula for the interior angle pentagon has five sides so we do five subtract 2 * 180 divide 5 this gives you the interior angle of 108 you might remember that one from before so both of those red angles are 108° now if mark on the interior angle of regular polygon so this angle here in green we can work this out because all three of those angles are angles around point and angles around our Point add up to 360° so if we do 360 and subtract both of those 108s we find the green angle is 144° in this question we're trying to work out the number of sides of regular poon the formula we had for the number of sides was 360 ID by the exterior but we've now found the interior angle of regular polygon so we need is exterior angle we can do that using this formula here remember the interior plus exterior must make 180 so if the interior is 144 the exterior must be 180 subtract this which gives you 36° we now know the exterior is 36 so we can use the sides formula the number of sides is 360 divide by the exterior which is 36 this gives you 10 so the shape must have 10 sides and that's the answer to this question let's try another example so in this question the diagram shows regular nonagon two equilateral triangles and regular polygon and we are asked to work out the number of sides that regular polygon has we're going to do this by working out these two angles here we'll start with the regular nonan the shape with nine sides to do this we'll need the interior angle formula since it has nine sides we do 9 subtract 2 multip 180 / 9 this gives you 140° and for the equilateral triangle it's quite easy we just need to do 180 divide 3 which gives you 60° so we can Mark both of these angles onto our diagram now we can work out this green angle here in the same way we did before the angles around point make 360 so if we do 360 subtract 140 and 60 we end up of 160° so we've now found the interior angle of regular polygon but the formula for the number of sides needs the exterior angle so we'll use this formula once again to find the exterior angle we know the interior one is 160 so if we do 180 subtract 160 we find the exterior was 20 we can now use the formula for the number of sides 360 IDE by the exterior which is 20 gives you the answer 18 so this shape had 18 sides thank you for watching this video hope you found it useful check out the one think you should watch next subscribe so you don't miss out on future videos and why not go and try the exam questions in this video's description