Step by step
The question
Question: find how many sides a regular polygon has, when its interior angle is 150°.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find the exterior angle
The interior angle is 150°.
The exterior angle sits next to it.
Together they make a straight line: 180°.
So the exterior angle is 180 − 150.
Careful: the 360° is for the exterior angles.
So we need the exterior angle first.
180 minus 150 is 30°.
2 Walk around the shape
Walk along the edge.
At a corner, carry the side on, dashed.
Then turn to the next side.
The turn is the exterior angle.
All the way round, we end up facing the same way as at the start.
That is one full turn: 360°.
So the exterior angles add up to 360°.
Regular means all the sides and all the angles are equal.
So it turns the same at every corner.
3 Count the turns
Each turn is 30°.
How many turns of 30° make 360°?
360/30.
Both are tens, so drop a zero from each: 36 divided by 3 is 12.
360 divided by 30 is 12.
So 12 turns.
There is one turn at each corner, so 12 corners and 12 sides.
Here is the whole shape: 12 sides.
4 Check
Let's check: 12 turns of 30°.
We get 360°, one full turn.
So 12 sides is right!
5 Pick the choice
Our answer is sides = 12.
That is choice B.
6 Why not the others
A is 6.
Wrong: 180 ÷ 30: the exterior angles add up to 360°.
C is 15.
Wrong: 150 ÷ 10: the sides come from the exterior angle.
D is 30.
Wrong: that is the exterior angle, not the number of sides.
Let's look back.
First: find the exterior angle.
Then: walk around the shape.
Then: count the turns.
Then: check.
Then: pick the choice.
Last: why not the others.