In the regular octagon below, find x.
The sum of the interior angles of an n-sided polygon is 180 • (n – 2)
In this octagon, that sum is 180 • (8 – 2) = 1,080 and since it's a regular octagon,
all the angles are equal, so one interior angle equals 135.
Look at angle x. It's the middle angle of three angles which total that 135.
If we could figure out the other two angles, angle x would be what's left over.
Call the angle to the left of x, call it y. And the angle to the right of x, call it z.
Angle y is one of four angles in a quadrilatereal.
The sum of the interior angles is 180 • (4 – 2) = 360.
The two big angles total 270 so that leaves 90 for the other two angles to total.
Since those other two angles are equal, each is 45, which of course means that y = 45.
Use the same reasoning to determine that angle z is half of (180 – 135), so thus z = 22.5.
Together, y + z = (45 + 22.5) = 67.5. Subtract that from 135 and it leaves x = 67.5.