Polygon angle sum via triangulation from a vertex

4.G.A.2 · adapt · grade 4

Archetype: Angle Facts in a Figure · step in a 13-type progression

In a regular nonagon (a regular polygon with 9 sides), find the measure of angle $a$ .

[Figure] A regular nonagon (9-sided regular polygon) is drawn, with one of its interior angles marked $a$ .

Show solution

Understand

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.

Givens

The polygon is regular (all 9 sides equal, all 9 interior angles equal).
It has 9 sides (a nonagon).
One interior angle is labeled a.

Unknowns

The measure of interior angle a.

Constraints

Drawing all diagonals from one vertex splits an n-sided polygon into (n - 2) triangles.
Each triangle's three angles add to 180 degrees.
Because the polygon is regular, all interior angles are equal, so each equals the total divided by 9.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram#9 Solve an Easier Related Problem

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.

Execute

#1 Draw a Diagram 4.G.A.2

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 \text{ triangles}

Fanning diagonals from one corner always makes two fewer triangles than the number of sides.

#9 Solve an Easier Related Problem 4.MD.C.7

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 \times 180^\circ = 1260^\circ

Seven triangles of 180 degrees combine into the polygon's full interior-angle total.

#7 Identify Subproblems 4.G.A.2

Because the nonagon is regular, all 9 interior angles are equal. Each one, including a, is 1260 / 9 = 140 degrees.

1260^\circ \div 9 = 140^\circ

Equal angles means just sharing the 1260 degrees fairly into 9 equal parts.

Answer: 140 degrees

Review

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.

Pattern approach (tool 5): each exterior angle of a regular 9-gon is 360 / 9 = 40 degrees, so each interior angle is 180 - 40 = 140 degrees -- the same answer.

Standards · min grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Triangulating the regular nonagon and using equal-angle property to split the total.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding the seven 180-degree triangles into the total interior-angle measure.

💡 Any polygon is just triangles in disguise -- 7 triangles of 180 degrees shared among 9 equal corners gives 140 degrees each!