Angles index

Description of Shapes and Angles

index --- angles --- right angle --- boat --- torch --- shape --- tree --- cannon --- logo

This page explores shapes and angles.

When children first meet angles, they can get confused by them. On the right are two angles. They are both the same size, but children may think that one is bigger than the other, just because the lines either side of the angle are longer. The problem is that the angle isn't really there; it's just defined by the lines. It's hard to think about something that isn't there!

It can be easier to think of a turn rather than an angle. If you face one way, then turn to face another way, then you have turned through an angle. You can even make a shape this way. Walk a few steps, turn through an angle, walk some more steps, turn again, and carry on doing this until you get back to where you started. If you are making a mark when you walk (possibly with muddy footprints!) then you end up with a shape. If the distances that you walk (the sides) and the turns (angles) are the same, then this will be a regular shape.

This webpage draws some regular shapes for you. It can be used in several different ways. You can select a number of sides, say 5. It will draw you the regular 5-sided shape, give its name (pentagon), and tell you what the internal and external angles are. If you prefer, you can select a name, such as a hexagon, and it will draw it for you, and tell you the number of sides and angles. So you could use the page as a kind of reference.

There are two types of angles for each shape, the internal angle and the external angle. Both these angles are important. The internal angle is the one normally associated with a shape. "The angles of a regular triangle are 60". This means the angle inside the shape. It's marked in pink on the left. When you draw a shape, you draw a straight line, mark the length of a side, then use a protractor to get the internal angle, and draw the next side.

The external angle is not the outside of the shape. If you extend (make longer) one side, it's the angle of that and the next side. It's marked in blue on the left. This may seem a weird thing to be interested in, but it's very helpful to know. If you are walking round a shape, when you get to a corner, you turn through the external angle, and walk along the next side. Since we are looking at turns to represent angles, then this is what we must think about. If you have a turtle (a robot which can be programmed with a language called Logo to move and turn while drawing a line), then you will need to turn it through external angles. The internal angle plus the external angle equals 180 or a straight line. Unfortunately, this means that the internal angle and external angle of a square are the same, 90, which can lead to confusion!

The external angle is easy to calculate for a regular shape. If you walk round a shape and end up where you started, and facing the same direction, then you will have turned through a complete circle, of 360. If the shape have 5 sides, then it has 5 angles, and these are the same size. So each angle is 360 / 5, or 72. If you want to then calculate the internal angle, you subtract the external angle from 180. So

Internal angle of regular polygon of n sides = 180 - 360 / n

There are various exercises that you can do with the webpage. The students could write down the internal and external angles for the simple shapes, and see if they could work out the relationship between them. Then they could see if they could spot how to calculate the external angle, possibly with a hint or two!

So far, I haven't mentioned the stars. These are constructed in exactly the same way as the simple shapes; you walk along a line, turn an angle, walk the same distance, turn the same angle, and so on, until you get back to where you started. The stars look different because they don't get back to where they started after one complete circle. Instead they need two or more circuits. This makes the lines cross each other, and give a star effect. This is rather attractive in its own right, but also is a good exercise for more able students. If you look at the angles, and select each one, one after each other, it can seem very confusing to work out why some make a simple shape, and others make a star. Make them think about it! They need to look at the external angles. Simple shapes have external angles which divide exactly into 360. The stars don't. Instead the stars' external angles divide into a multiple of 360. For example, the Five Star has external angle of 144, which divides into 2 x 360, or 720.

You may notice that there is no seven sided shape, or eleven sided shape. I wanted to keep whole angles, and since 7 and 11 do not divide exactly into 360, their external angles, and hence their internal angles, will have fractions in them. They do exist, it's just that they're messy!

The square perhaps needs a mention. First, it's a strange way up. You think of squares being flat on one of their sides, and this one is balanced on a point. But all the other shapes start at the top point, so it seemed better to make the square the same. Anyway, it's a good idea to realise that a square is a square, even when it doesn't look quite right! The square is also different, as it is always a regular shape. The other shapes have irregular forms, so strictly speaking, you should say a regular triangle, a regular pentagon, and so on. The general name for a four sided shape is a quadrilateral. There are other special quadrilaterals, such as rectangles, parallelograms and rhomboids.

I have used "shape" as a description. Strictly speaking, they are called polygons. Precision of language is important in Mathematics, but it's not a good idea to overwhelm children with words unnecessarily.

Click here for the shapes and angles webpage.