How many degrees is the gap angle x?
A normal-level "Angle" problem.Equilateral triangles, squares and regular hexagons tile the plane; regular pentagons cannot — 108° does not divide 360° exactly.
GivenTwo regular pentagons placed edge to edge
Hints
- Look around the gap. At this one point there are three things: two pentagon corners and the gap x.
- First, one corner of a regular pentagon. A pentagon splits into three triangles, so its angles total 180 × 3 = 540°. All five are equal: 540 ÷ 5 = 108°.
- Two pentagon corners come to 108 + 108 = 216°. Those two take up part of the full turn at this point.
- Here is the rule we need: going right round a point is 360°. Including the gap, these three make exactly one full turn.
- So the gap left over is 360 − 216.
Answer 144°
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