You already know that the three angles of a triangle add up to
180^\circ. That single fact unlocks every polygon: a
four-sided shape, a five-sided shape, a hundred-sided shape. The trick is to cut the polygon
into triangles and count.
Pick one corner of an n-sided polygon and draw a diagonal to each
of the other corners. The polygon falls apart into exactly
n - 2 triangles, and the polygon's interior angles are just the
triangles' angles gathered up:
\text{interior angle sum} = (n - 2) \times 180^\circ
For any polygon with n sides:
-
diagonals from one corner split it into n - 2 triangles;
-
each triangle contributes 180^\circ by the
triangle angle sum;
-
so the interior angles add up to
(n - 2) \times 180^\circ;
-
for example a quadrilateral makes 360^\circ, a pentagon
540^\circ, and a hexagon 720^\circ.
Why it works
Watch a pentagon (five sides) come apart. From a single corner, two diagonals carve it into
three triangles — and 5 - 2 = 3, exactly as the rule predicts.
Every interior angle of the pentagon is built from angles of those triangles, with nothing
left over and nothing double-counted. Three triangles, each worth
180^\circ, give
3 \times 180^\circ = 540^\circ. The same cut works for any polygon:
an n-gon always splits into n - 2
triangles.