Why does the sum of the angles in a triangle equal 180 degrees?

Answer 1

The sum of the internal angles of a triangle, indeed of any polygon, is always a constant entirely dependent upon the number of angles required to circumscribe, with straight lines, the shape of a two-dimensional (plane) figure.

To explain and prove why this is so we'll start with a regular polygon.

NOTE: At this stage you may want to print this page so that you can more easily follow the steps and simple calculations.

1.1) For a regular polygon, each side is the same length and all the internal angles are equal, and all the external angles are equal.

1.2) If you have a long piece of wire and you want to bend it at regular intervals (to make a polygon) so that when you get to the end you will make the last section of the wire lie on top of, and parallel with, the first section of the wire, then you will have to bend the wire in a full circle of rotation. A full circle is a rotation through 360 degrees of movement. (Keep 360oin mind!)

1.3) So if you want to do this in twelve equal-angled bends, each bend will be at 1/12th of a full circle. 1/12th of 360o is 30o. So each bend will be at 30o to the straightness of previous section. We will call this angle the external angle.

1.4) In this example, even if you make a mistake and bend one of the angles at say 31o rather than 30o you will have to bend another angle differently to compensate. In this case you will have to do another of the bends at a 29o angle, and only then can the full circle of rotation of the end of the wire be completed.

1.5) In fact, regardless of the number of bends you choose to make, you will always have to have rotated the last section of the wire in a full circle (360o) from the wire's starting position.

• Therefore, as the line is going all the way around the shape, the sum of the external angles of all the bends will always be 360o, i.e. one complete revolution.

2.) Now some examples of this 'external angle' rule

1. To bend some wire into a regular three sided figure, each external angle needs to be 360 divided by 3, i.e. 120o. (Or, to put it another way, 3 x 120o = 360o, a full circle.) Even for an 'irregular' three sided shape, the sum of all the external angles must be 360o in order to complete the full circle of rotation.
2. Similarly, for a 4-sided polygon the four external angles must average 90o (360 divided by 4) for the the sum of the external angles to equal 360o.
3. What would be the average of the external angles of a 5-sided polygon? Answer: 360 divided by 5 = 72o. (5 x 72o = 360o)

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3.) Now let's see the consequences on the internalangles.

3.1) Consider a horizontal piece of wire. The right half of its length is at an angle of 180o to the left half. If you bend the wire up slightly, at say 10o from the horizontal, then the 'internal' angle, between the left half and the right half, will be 170o i.e. 180o- 10o.

3.2) The internal angle of a deviation from the straight line will always be at an angle that is "180 degrees, lessthe external angle".

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4.) From these facts we can work backwards and devise a formula which will tell us what the sum, and the average, of the internal angles of any regular polygon will be!

Let

n = the number of angles (or sides)

e = the average external angle of variation

x = the average internal angle.

T = the sum Total of all the internal angles.

Formula:

360o divided by n = e

180o minus e = x

x times n = T

Some examples:

4.1) Let's try this formula with the triangle, which has 3 angles (or sides).

360o divided by 3 = 120o

180o - 120o = 60o. So the average internal angle will be 60o

and 3 x 60o = 180o, the sum total of all the internal angles.

4.2) Now try the formula with a 4-sided figure:

360o / 4 = 90o

180o - 90o = 90o. So the average of each internal angle will be 90o, and the total will be

4 x 90o = 360o.

4.3) What about a five sided polygon?

360o / 5 = 72o

180o - 72o = 108o, the average for each internal angle.

5 x 108o = 540o, the sum total of all its interior angles.

4.4) So we have seen that the internal angles (average or sum Total) for polygons with any number of sides can be calculated using the above formula.

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Interesting experiments:

5.1) Try the formula, as set out above, with a polygon having two sides! Although 2-sides is physically impossible, and it is an 'imaginary polygon'(!) What does an imaginary polygon look like?! nevertheless you will find that the formula holds true!

5.2) Now try the formula with a imaginary polygon with ONE side! You will be surprised! Not only does it work, but you will see how the formula is totally reliable, even with imaginary polygons!

Rule, and reason: The sum of all the externalangles (as defined above) of a polygon will always be 360o ... because the line has to make a full circle 'rotational movement' in order to join up again and make the complete shape.

Even though the overall shape is not a circle, the sides/lines must progressively go round in a completely circular movement (a total of 360o to the horizontal) to join up and complete the polygon.

The sum of the corresponding reciprocal internal angleswill also always be the same for all polygons which have the same number of sides.

For diagrams and more information, see Related linksbelow.