Angles in a polygon, step by step
Every polygon's angles add to a number you can work out rather than look up.
Angles in a 10-sided shape
- There is no separate rule for each shape. Pick one corner of the 10-sided shape and draw a line from it to every corner it is not already joined to — the whole shape falls into triangles, and you already know what a triangle's angles come to.
- Counting them: the corner you started from reaches every other corner except its own two neighbors, so a 10-sided shape splits into 10 − 2 = 8 triangles.
- Every triangle's angles add to 180°, and between them the triangles use up every angle of the shape and nothing else. So the whole lot comes to 8 × 180 = 1440°.
- Now the question wants ONE angle, and it may be divided only because this shape is regular — every angle the same. 1440 shared between 10 is 1440 ÷ 10 = 144°.
- So each angle of a regular 10-sided shape is 144°.
How it works.
- 01
There is no separate rule for each shape. Pick one corner of the 10-sided shape and draw a line from it to every corner it is not already joined to — the whole shape falls into triangles, and you already know what a triangle's angles come to.
- 02
Counting them: the corner you started from reaches every other corner except its own two neighbors, so a 10-sided shape splits into 10 − 2 = 8 triangles.
- 03
Every triangle's angles add to 180°, and between them the triangles use up every angle of the shape and nothing else. So the whole lot comes to 8 × 180 = 1440°.
- 04
Now the question wants ONE angle, and it may be divided only because this shape is regular — every angle the same. 1440 shared between 10 is 1440 ÷ 10 = 144°.
- 05
So each angle of a regular 10-sided shape is 144°.