Angles

We talk about angles in relation to a point of view, the direction someone might be coming from, a form of fishing (angling) and/or the focus of a new story. This makes 'angle' an interesting word to investigate and focus on in terms of its mathematical meaning, and consider how this relates to its other definitions.

Sometimes children think of angles as static. They might only look at the pointy part, or at the arms and think of these as the 'angle'. A common alternate conception is: the longer the arms, the greater the angle. These parts - the arms and the vertex or corner (pointy part) - are important, but considered in isolation they can distract from a full understanding of what an angle is.

Angles integrate both geometric and measurement concepts.

An angle is a measure of turn.
An angle is created when two lines meet at a common point.
Angles are described between 0 and 360 degrees. This is a full rotation.

Looking around you now, what is the most common angle you can see?

We will take a guess and think it is probably 90 degrees if you are sitting in a building or near built structures. This is an interesting talking point - why are there a lot of 90 degree angles in our buildings?

You may also notice angles as the hands of a clock turn or how a door sits when it is opened. Your arms and legs make different angles depending on what you are doing - sitting, standing, holding a ball, using a computer, or skating down a hill. The angles we use and create are purposeful, making them very interesting to explore - with our bodies, with materials, in the environment, in art and the arts... there are so many applications and opportunities.

The easiest angles to learn about are those with two arms (like the two hands of the clock above). This makes the angle - or measure of turn between the two arms - easy to see. Two-arm angles are often seen in two-dimensional shapes (see 2D Shapes), and are most obvious in the corner of windows, doors, the edges of books etc.

Angles can be described as interior and exterior.

Interior angles are inside a shape. Exterior angles are outside a shape.

Angles can be named based on their measure of turn. This includes fractional descriptions, for example 1/4 and 1/2 turns. These fractions relate to the 360 degree whole. So a quarter turn is 1/4 of 360 degrees which is 90 degrees.

They are also named as a 'type' of angle. The following table shows you these types and their ranges. Remember they are all related to the full rotation or revolution, i.e. 360 degrees.

Using what you know to work out what you don't know is a powerful mathematical strategy. If we consider two-arm angles inside a 2D shape called a polygon (see 2D Shapes), we can work out unknown angles based on those we do know. To help us out with this, we can keep in mind that finding the sum of the angles inside regular and irregular polygons is the same. For example, the triangle in the left of the picture below is called a right-angled triangle because one of its 3 angles is 90 degrees. It is an irregular polygon (see 2D Shapes).

The triangle at the right of the picture below is an equilateral triangle as all the sides are equal. All the angles are also equal. They are all 60 degrees.

You will notice if you add up the angles in the irregular (right angled) triangle the sum is:

90 + 60 + 30 = 180 degrees.

If you add up the angles in the regular (equilateral) triangle the sum is:

60 + 60 + 60 = 180 degrees.

As mathematics is the study of patterns and relationships, we can use this information about the sum of the internal angles of triangles to help us work out the angles of other regular and irregular polygons.

Have a look at the table below - what patterns do you notice?

What relationships can you see?

You may have noticed that 180 degrees features in all of the sum of internal angles column. It might be helpful for you to draw a rectangle or a pentagon and divide it internally into triangles from a common point.

What do you notice?

You may have divided the rectangle into 2 triangles. This means the sum of the angles in a rectangle is 2 x 180 degrees.

You may then have divided the pentagon into 3 triangles. This means the sum of the angles in a pentagon is 3 x 180 degrees. This is helpful!

As you continue looking for patterns, you may notice a relationship between the number of triangles you could create from a common point and the number of sides in that shape... does this always work?