Step by step
The question
Question: find one interior angle of a regular octagon.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Cut into triangles
An octagon has 8 sides and 8 corners.
A polygon is a flat shape with straight sides.
Interior angles are the angles inside, at the corners.
Regular means all the sides are equal, and all the angles are equal.
Pick one corner.
Draw lines from it to every corner that is not next to it.
They cut the octagon into triangles.
Count them: 1, 2, 3, 4, 5, 6.
Each triangle sits on one side of the octagon, away from our corner.
The two sides that touch our corner have no triangle of their own.
So there are 2 fewer triangles than sides: 8 − 2 = 6.
2 Add the triangles
The angles in each triangle add up to 180°.
The corners of the triangles fill the corners of the octagon exactly.
So together they make all its angles.
6 triangles, 180° each: 6 × 180.
180 is 18 tens.
So forget the zero: 6 times 18 is 108.
Put the zero back: 1080.
So the angles add up to 1080°.
3 Share it out
All 8 angles are the same.
So we share 1080° equally: 1080/8.
Split 1080 into 800 and 280.
800 divided by 8 is 100.
280 divided by 8 is 35.
Together 135.
So each angle is 135°.
4 Check
Let's check: 8 angles of 135° together.
We get 1080°, the angle sum.
So 135° is right!
5 Pick the choice
Our answer is one angle = 135°.
That is choice B.
6 Why not the others
A is 45°.
Wrong: that is the exterior angle.
C is 157.5°.
Wrong: used 8 − 1: the sum is (8 − 2) × 180°.
D is 1080°.
Wrong: that is the sum of all 8 angles.
Let's look back.
First: cut into triangles.
Then: add the triangles.
Then: share it out.
Then: check.
Then: pick the choice.
Last: why not the others.