Close a chain of segments into a polygon — and two angle laws fall out for free.
Three sticks fence a triangle. Add a fourth stick and you fence a quadrilateral; a fifth, a pentagon; keep going and you can fence any polygon you like. Each one is nothing more than a closed loop of straight sides — yet the very instant you close the loop, the angles inside stop being free to do whatever they please. Slice the figure with a few diagonals and it falls into a tidy pile of triangles you already understand. Walk once around its rim and you turn through exactly one full circle. Two laws — one for the inside angles, one for the outside turns — and every polygon on Earth obeys them.
A polygon is a closed figure built from straight sides that meet only at their endpoints. Those meeting points are the vertices (one is a vertex). A side and a vertex are simple ideas, but they come with one strict rule: the sides may touch only at vertices — no crossing, no dangling ends. Close up three sides and you have a triangle; four, a quadrilateral; and so on.
A segment that joins two vertices which are not next to each other is a diagonal. (A side joins adjacent vertices; a diagonal skips across the inside.) We name a polygon by its number of sides:
| Sides | 3 | 4 | 5 | 6 | 7 | 8 | n |
|---|---|---|---|---|---|---|---|
| Name | triangle | quadrilateral | pentagon | hexagon | heptagon | octagon | n-gon |
How many diagonals start at a single vertex? From any vertex you cannot draw a diagonal to itself, nor to its two neighbours, so it reaches the other (n−3) vertices. That gives (n−3) diagonals from one corner. Counting all the diagonals, every one is shared by two vertices, so the polygon has
n(n−3) ⁄ 2 diagonals in all.
For the hexagon above: (6−3) = 3 from each vertex, and 6·3⁄2 = 9 diagonals altogether.
From one vertex of an n-gon you can draw (n−3) diagonals. The whole polygon has n(n−3)⁄2 of them.
Stretch a rubber band around the corners of a polygon. If the band hugs every vertex and never has to cave inward, the polygon is convex: every diagonal stays inside it, and every interior angle is less than 180°. If the band would have to dent inward to reach a "tucked-in" vertex, the polygon is concave — it has at least one interior angle greater than 180° (a reflex angle), and at least one diagonal pokes outside.
A polygon is regular when it has both all sides equal and all angles equal. The equilateral triangle, the square, the regular pentagon, the regular hexagon — these are the regular polygons. The tick marks below show equal sides; matching arcs show equal angles.
"Regular" needs both conditions. Equal sides alone is not enough — a rhombus has four equal sides but slanted angles. Equal angles alone is not enough either — a rectangle has four right angles but two long sides and two short. Only when sides and angles are all equal is the polygon regular (a square is the regular quadrilateral).
Here is the first law, and it is pure triangle bookkeeping. Pick one vertex and fan out every diagonal you can from it. A triangle gives no diagonals — it is already one triangle. A quadrilateral splits into 2 triangles, a pentagon into 3, a hexagon into 4. Each new side adds exactly one more triangle, so an n-gon splits into (n−2) triangles.
Every triangle holds 180° of angle, and the diagonals don't add or remove any angle from the corners — they only carve the inside into pieces. So the polygon's interior angles must total the triangles' angles:
interior-angle sum = (n−2)·180°
A pentagon: (5−2)·180° = 3·180° = 540°. For a regular n-gon all n angles are equal, so each one is the total shared out evenly:
each interior angle = (n−2)·180°n
A regular pentagon: 540° ÷ 5 = 108° each. Slide the stepper and watch the triangles — and the running total — appear.
An octagon has 8 sides. Interior-angle sum = (8−2)·180° = 6·180° = 1080°. If it is regular, each angle = 1080° ÷ 8 = 135°.
Now the second law, and it has a beautiful walk-around picture. At each vertex, extend one side a little past the corner. The angle between that extension and the next side is the exterior angle — it is the amount you turn as you walk along the rim and round that corner. At every vertex,
interior angle + exterior angle = 180°
because the side and its extension form a straight line. Now take the walk. Start along one side and stroll all the way around the polygon, turning by the exterior angle at each corner, until you face your starting direction again. You have made exactly one full turn. So:
the exterior angles of any convex polygon sum to 360°
— and this holds for every convex polygon, no matter how many sides! A triangle, a hexagon, a 100-gon: one lap around the outside is always one complete turn. For a regular n-gon the turns are all equal, so each exterior angle = 360°⁄n.
Interior angles depend on n: they sum to (n−2)·180° and grow with more sides. Exterior angles do not: they always sum to 360°. At every vertex the two add to 180°, which is why the same picture gives both.
Why does this matter to a tiler? When identical tiles cover a floor with no gaps and no overlaps, the corners that crowd around any meeting point must fit together perfectly — their angles must add to exactly 360°, a full turn with nothing left over and nothing doubled. This corner-fitting is what makes a tessellation work.
For a regular tile, that means its interior angle must divide 360° evenly. Only three regular polygons pass the test:
| Regular tile | Interior angle | Around a point | Tiles? |
|---|---|---|---|
| triangle | 60° | 60° × 6 = 360° | yes ✓ |
| square | 90° | 90° × 4 = 360° | yes ✓ |
| pentagon | 108° | 108° × 3 = 324° | no ✗ |
| hexagon | 120° | 120° × 3 = 360° | yes ✓ |
The regular pentagon is the famous failure: three of them leave a 36° gap (3·108° = 324°), and four overlap (4·108° = 432°). No whole number of regular pentagons closes a corner. Try each tile and watch the angles meet — or fail to meet — at the center.
• A polygon is a closed figure of straight sides meeting only at vertices; a diagonal joins two non-adjacent vertices.
• From one vertex: (n−3) diagonals; in all n(n−3)⁄2.
• Convex = never caves in (all angles < 180°). Regular = all sides equal and all angles equal.
• Interior-angle sum = (n−2)·180° (an n-gon = (n−2) triangles). Regular: each = (n−2)·180°⁄n.
• Exterior-angle sum = 360° for every convex polygon (one lap = one full turn). Regular: each = 360°⁄n.
• A regular tile tessellates only when its interior angle divides 360°: triangle, square, hexagon — not the pentagon.
What is a 7-sided polygon called, and what is the sum of its interior angles?
A heptagon. Interior-angle sum = (7−2)·180° = 5·180° = 900°.
Find the sum of the interior angles of an octagon (8 sides).
(8−2)·180° = 6·180° = 1080°.
What is each interior angle of a regular hexagon?
Sum = (6−2)·180° = 720°; shared by 6 equal angles, so each = 720° ÷ 6 = 120°.
Each exterior angle of a regular polygon is 40°. How many sides does it have?
The exterior angles total 360°, and they are all equal, so n = 360° ÷ 40° = 9 sides (a regular nonagon).
A convex polygon has 12 sides. What is the sum of its exterior angles?
360° — the exterior angles of any convex polygon sum to 360°, no matter how many sides it has. (The "12" is a distraction.)
Which regular polygons can tile a floor by themselves, and why?
The equilateral triangle (60°), the square (90°), and the regular hexagon (120°) — because each interior angle divides 360° exactly (360°⁄60 = 6, 360°⁄90 = 4, 360°⁄120 = 3). A regular pentagon's 108° does not divide 360°, so it leaves a gap.
Six questions to lock it in. Tap the answer you think is right.
This lesson opens Stage 16 by turning the triangle's angle sum — already familiar — into a tool that measures any polygon. The big idea is decomposition: fan diagonals from one vertex and an n-gon becomes (n−2) triangles, which makes the formula (n−2)·180° feel inevitable rather than memorized. The exterior-angle law is even cleaner: one walk around the rim is one full turn, so the exterior angles always sum to 360°, independent of n. Tessellation then puts both ideas to work, asking which regular tiles fit 360° at a corner.
The misconception to watch. The word "regular" is the trap. Students reliably hear it as "equal sides" and forget that the angles must be equal too — so they wrongly call a rhombus or a rectangle "regular." Keep the two non-examples handy: a rhombus has four equal sides but unequal angles; a rectangle has four equal angles but unequal sides. Only when both hold is the polygon regular. A second slip: applying the (n−2)·180° formula to a concave figure by miscounting the triangles when a diagonal falls outside — which is exactly why we restrict the clean statements to convex polygons.
Common Core. The interior- and exterior-angle reasoning is 8.G.A.5 (informal arguments about angle sums) extended toward the high-school standards HS G-CO.C.11 and G-MG (modeling with geometric figures); tiling connects to 7.G and the rigid-motion strand of G-CO. Throughout, encourage the habit this stage rewards: state the rule out loud, then write the one-line conclusion.