Problem
Polygon angle sum via triangulation from a vertex
I have a regular nonagon -- a 9-sided polygon with all sides and all angles equal. One interior angle is marked a, and I need its measure in degrees.
Geometry
Your answer
degrees
Watch out Stopping at 1260 degrees gives the sum of all nine angles, not the size of just one -- you still need to divide by 9.
How to solve
Strategy Identify Subproblems — Triangulate the nonagon from one vertex to turn an unfamiliar 9-gon into a familiar set of triangles, add up the angles using 180 degrees per triangle, then split equally because the polygon is regular.
1STEP 1
Cut the nonagon into triangles
From a single vertex, draw diagonals to the non-adjacent vertices. A polygon with 9 sides breaks into 9 - 2 = 7 triangles.
9 - 2 = 7 triangles
Fanning diagonals from one corner always makes two fewer triangles than the number of sides.
4.G.A.2Draw A Diagram2STEP 2
Add up all the interior angles
Each triangle contributes 180 degrees, and these triangle angles together fill exactly the nonagon's 9 interior angles. So the total of all interior angles is 7 x 180 = 1260 degrees.
7 × 180° = 1260°
Seven triangles of 180 degrees combine into the polygon's full interior-angle total.
4.MD.C.7Solve An Easier Related ProblemThe nine interior angles of the regular nonagon add up to 1260 degrees altogether.
Why?
The nonagon splits into seven triangles that fill it completely, and adding up those seven triangles' angles gives the very same amount as adding up the nonagon's own interior angles.
Why?
The diagonals drawn from one corner cut the nonagon into triangular pieces with no gap between them and no overlap, so all the angle inside the pieces adds back to exactly the angle inside the whole nonagon.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
The nine sides split into the two that meet at the chosen corner and the seven that do not, and each of those seven far sides is the base of one triangle, so there are exactly seven triangles.
Why?
Each triangle rests on exactly one far side and each far side is the base of exactly one triangle, so the triangles and the far sides must be equal in number.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Each of the seven triangles holds 180 degrees, and seven equal amounts of 180 degrees is 7 times 180, which is 1260.
Why?
The three corners of any flat triangle always meet to make one straight angle of 180 degrees.
🧱A triangle's angles sum to 180 degreesThe three corners of any flat triangle always add up to one straight angle, 180 degrees.
Why?
Seven triangles that each carry 180 degrees are seven equal groups of 180, and combining equal groups is exactly what multiplying does.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
3STEP 3
Divide equally among the 9 angles
Because the nonagon is regular, all 9 interior angles are equal. Each one, including a, is 1260 / 9 = 140 degrees.
1260° ÷ 9 = 140°
Equal angles means just sharing the 1260 degrees fairly into 9 equal parts.
4.G.A.2Identify SubproblemsAnswer
140 degrees
1260° ÷ 9 = 140°
140 degrees is between 90 and 180 degrees, which fits a convex regular polygon (each angle is obtuse but less than a straight line). As polygons gain sides their angles grow toward 180, and 140 for a 9-gon sits sensibly between a hexagon's 120 and larger polygons.
Takeaway
Any polygon is just triangles in disguise -- 7 triangles of 180 degrees shared among 9 equal corners gives 140 degrees each!
- Cut the nonagon into triangles
- Add up all the interior angles
- Divide equally among the 9 angles
Where next?
Another one like thissuggested
Step back and solidifysuggested