Exterior angle sum of a regular polygon is 360

4.G.A.24.MD.C.7 · adapt · grade 4

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

Each side of a regular hexagon is extended, as shown. Find the sum of the measures of angles $a$ , $b$ , $c$ , $d$ , $e$ , and $f$ .

[Figure] Each of the six sides of a regular hexagon is extended in one direction, so that at each vertex an exterior angle is formed between a side and the extension of the next side. The six exterior angles are labeled $a$ , $b$ , $c$ , $d$ , $e$ , $f$ in order around the hexagon.

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Understand

A regular hexagon has each of its six sides extended in one direction. At every vertex this makes one exterior angle between a side and the extension of the next side. I need to add up all six exterior angles a, b, c, d, e, f.

Givens

The polygon is a regular hexagon (6 equal sides, 6 equal interior angles).
Each side is extended in one direction, forming one exterior angle at each of the 6 vertices.
The six exterior angles are labeled a, b, c, d, e, f around the hexagon.

Unknowns

The sum a + b + c + d + e + f of the six exterior angles.

Constraints

At each vertex, the interior angle and its exterior angle lie on a straight line, so they add to 180 degrees.
Since the hexagon is regular, all six interior angles are equal and all six exterior angles are equal.

Plan

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

Break the total into two easy facts: the interior angles of a hexagon add to a known amount, and at every vertex interior + exterior = 180 because a straight line is 180 degrees. Combining these gives the exterior-angle sum without measuring anything.

Execute

#9 Solve an Easier Related Problem 4.G.A.2

Split the hexagon into 4 triangles by drawing diagonals from one vertex; the 6 interior angles add to 4 x 180 = 720 degrees. Because the hexagon is regular, each interior angle is 720 / 6 = 120 degrees.

\frac{(6-2)\times 180^\circ}{6} = \frac{720^\circ}{6} = 120^\circ

Cutting a polygon into triangles is exactly the Grade 4 way to find angle sums; each triangle is 180 degrees.

#1 Draw a Diagram 4.MD.C.7

A side and the extension of the next side form a straight line at the vertex, so the interior angle and the exterior angle together make 180 degrees. Each exterior angle is therefore 180 - 120 = 60 degrees.

180^\circ - 120^\circ = 60^\circ

On a straight line the two angles must fill 180 degrees, so subtraction gives the leftover exterior angle.

#7 Identify Subproblems 4.MD.C.7

All six exterior angles are equal to 60 degrees, so the sum is 6 x 60 = 360 degrees.

6 \times 60^\circ = 360^\circ

Adding six equal angles is just repeated addition (multiplication), a skill young learners already have.

Answer: 360 degrees

Review

Six angles of 60 degrees each is 360 degrees, which is exactly one full turn. That makes sense: if you walked around the outside of the hexagon turning by each exterior angle, you would end up facing your original direction after one complete loop.

Use the pattern (tool 5): the exterior angles of ANY convex polygon always add to 360 degrees, no matter how many sides. Checking with the hexagon (6 x 60 = 360) confirms the rule.

Standards · min grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Recognizing the regular hexagon and finding its interior angle by splitting it into triangles.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using interior + exterior = 180 on a straight line and adding the six equal exterior angles.

💡 Walking once around any polygon turns you a full circle, so the outside angles always add up to 360 degrees!