What counts as a polygon
You have spent the last rung living inside triangles. A polygon is the natural next step: a closed figure built from straight segments joined end to end, with each corner touching exactly two of them and no two sides crossing in the middle. A triangle is the smallest polygon, with three sides; a four-sided one is a quadrilateral, then come the pentagon (5), hexagon (6), heptagon (7), octagon (8), and on upward. The segments are the sides and the corner points are the vertices; an n-sided polygon has n of each.
One distinction shapes everything that follows: a polygon is either convex or concave. In a convex polygon every interior angle is less than 180 degrees, so no corner caves inward and any segment drawn between two of its points stays entirely inside. A concave (or 're-entrant') polygon has at least one corner that pokes inward, like a dented arrow or a star. Unless we say otherwise, the polygons in this guide are convex — that is what keeps the angle-sum reasoning clean.
Cutting a polygon into triangles
We already own one golden fact from the triangles rung: the three interior angles of any triangle add to exactly 180 degrees — the triangle angle sum. The trick for every other polygon is to refuse to learn anything new and instead chop the shape into triangles. A diagonal is a segment joining two non-adjacent vertices. Pick one vertex of the polygon and draw every diagonal from it; the whole interior falls apart into a fan of triangles, with nothing left over and nothing overlapping.
Now just count the triangles in the fan. From your chosen vertex you cannot draw a diagonal to itself or to its two neighbours, so the diagonals reach the other n - 3 vertices; those n - 3 diagonals slice the interior into n - 2 triangles. A quadrilateral (n = 4) splits into 2 triangles; a pentagon (n = 5) into 3; a hexagon (n = 6) into 4. The pattern 'two fewer than the number of sides' is the heart of the whole formula.
Every interior angle of the polygon is built up entirely from the corners of these triangles, and no triangle-corner is wasted on anything else. So the polygon's interior angles add up to exactly (number of triangles) times 180 degrees. That is the interior angle sum: for a convex polygon with n sides, the sum S = (n - 2) * 180 degrees. Check it on the triangle itself, n = 3: (3 - 2) * 180 = 180, exactly the fact we started from.
The formula at work
With S = (n - 2) * 180 in hand, the angle sum of any polygon is a quick substitution. A pentagon's five angles add to (5 - 2) * 180 = 540 degrees; a hexagon's to (6 - 2) * 180 = 720; an octagon's to (8 - 2) * 180 = 1080. The formula also runs backward. If a polygon's interior angles add to 1440 degrees, solve (n - 2) * 180 = 1440, so n - 2 = 8 and n = 10 — a ten-sided decagon.
n name triangles (n-2) interior sum (n-2)*180 3 triangle 1 180 4 quadrilateral 2 360 5 pentagon 3 540 6 hexagon 4 720 8 octagon 6 1080 10 decagon 8 1440
Pause on the n = 4 row: a quadrilateral's four interior angles always add to 360 degrees. That single number governs the entire quadrilateral family you meet in the next guide — squares, rectangles, parallelograms, kites and the rest all share this 360-degree budget, and many of their properties are just clever ways of spending it. Worth filing away now.
Regular polygons and equal angles
A regular polygon is one that is both equilateral (all sides equal in length) and equiangular (all angles equal in measure). Both conditions are needed: a rhombus has equal sides but unequal angles, while a rectangle has equal angles but unequal sides, so neither is regular — only the square, which is both, qualifies among four-sided shapes. The equilateral triangle, the square, the regular pentagon, and the regular hexagon are the everyday faces of this family.
Because all n angles of a regular polygon are equal, finding one is just sharing the total fairly. Take the interior sum (n - 2) * 180 and divide by n: each interior angle measures (n - 2) * 180 / n degrees. A square gives (4 - 2) * 180 / 4 = 90, exactly as expected. A regular pentagon gives 540 / 5 = 108. A regular hexagon gives 720 / 6 = 120 — which is precisely why hexagons tile a honeycomb so neatly, since three 120-degree corners meet around a point to fill the full 360.
- Find one interior angle of a regular octagon (a stop-sign shape), so n = 8.
- Interior angle sum: (8 - 2) * 180 = 6 * 180 = 1080 degrees.
- Share it among the 8 equal angles: 1080 / 8 = 135 degrees per corner.
- Sanity check: 135 is less than 180, so the octagon is convex, as it should be.
The exterior angles: a constant 360
There is a second way to measure a polygon's corners that is, surprisingly, even simpler. At each vertex, extend one side past the corner; the angle between that extension and the next side is an exterior angle. An interior angle and its exterior angle at the same vertex sit along a straight line, so they form a linear pair and add to 180 degrees — exactly the linear-pair relation you proved back in the triangles rung.
Now imagine walking once around the outside of the polygon, returning to your start facing the original direction. At each corner you turn by exactly that vertex's exterior angle, and by the end you have turned through one full revolution. So for any convex polygon, the exterior angles add to 360 degrees — and remarkably, this total does not depend on n at all. A triangle, a hexagon, a hundred-sided shape: one trip around, one full 360-degree turn.
For a regular polygon this hands you the angles almost for free. Since the n exterior angles are all equal and add to 360, each one is 360 / n degrees. A regular pentagon has exterior angles of 360 / 5 = 72, so each interior angle is 180 - 72 = 108 — matching what we found the long way. Many problems are fastest through the exterior door: to find how many sides a regular polygon has when each exterior angle is 30 degrees, just compute n = 360 / 30 = 12.
Where this rung is heading
You now hold two formulas earned, not memorized: interior angles sum to (n - 2) * 180, and exterior angles always sum to 360. Both grew from a single seed — cutting the shape into triangles — so they are really the triangle angle sum in disguise, applied n - 2 times. That habit of reducing the unfamiliar to triangles is the deepest move in plane geometry, and it will keep paying off.
An honest caveat to carry forward: the interior-sum formula (n - 2) * 180 holds for convex polygons exactly as stated, and it still gives the correct total for concave polygons too, but the per-vertex story gets subtler when a corner reflexes past 180 degrees. The clean exterior-angle 'one full turn equals 360' picture, in particular, assumes you keep turning the same way — convex is where the intuition is unclouded, which is why we stay there.
Next we zoom in on the four-sided case. The 360-degree budget you uncovered here becomes the backbone of the quadrilateral family tree — how parallelograms, rectangles, rhombuses, squares, trapezoids, and kites nest inside one another, and which angle and side properties each one inherits. After that, the rung leaves straight edges behind entirely and steps into the circle, where these same counting instincts reappear as the central and inscribed angle theorems.