Generalizing Polygon Patterns
Use patterns to generalize properties of polygons, including exterior angles.
Students generalize polygon exterior-angle patterns by recognizing a constant total turn, explaining why regular polygons have exterior angle 360°/n, and distinguishing inverse variation from constant subtraction.
Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.
Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.
Learn the concept
A complete trip makes one full turn
For a convex polygon, take one exterior angle at each vertex while traveling around the boundary in the same direction. The total turn is 360°, regardless of the number of sides.
Regular polygons share the total equally
A regular polygon has congruent exterior angles. If n is the number of sides and e is each exterior-angle measure, then , so and .
Recognize the invariant
Side count and exterior-angle measure have a constant product, not a constant difference. Doubling n halves e. Each interior angle is supplementary to its adjacent exterior angle, so its measure is .
Worked example
Compare the exterior angles of regular polygons with 8, 12, and 24 sides. Then find each interior angle of the 24-sided polygon.
- For 8 sides, e = 360° ÷ 8 = 45°.
- For 12 sides, e = 360° ÷ 12 = 30°.
- For 24 sides, e = 360° ÷ 24 = 15°. Each side-count-and-angle product is 360°.
- The 24-sided polygon's interior angle is 180° − 15° = 165°.
Exterior angles: 45°, 30°, and 15°; 24-sided interior angle: 165°
Watch the lesson
Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.
CA Geometry: Exterior angles | Worked examples | Geometry | Khan Academy
Khan Academy · Geometry (TX TEKS) courseUse the full-turn exterior-angle sum to reason about polygon angle measures.
Watch on YouTubeTry it yourself
TEKSForge practice · unscored
1. A regular polygon has an exterior angle of 40°. Find its number of sides.
Show answer and reasoning
9 sides
Divide the 360° total turn by 40° per vertex: 360 ÷ 40 = 9.
2. A student says every exterior angle gets 10° smaller whenever a regular polygon gains one side. Test the claim using 6, 7, and 8 sides.
Show answer and reasoning
The claim is false.
The measures are 60°, about 51.43°, and 45°. The decreases are about 8.57° and 6.43°, not a constant 10°.
Released STAAR practice
Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.
Archived released STAAR Geometry EOC · 2013 · Item 19 · 1 point
The table contains a pattern formed by the number of sides and the measure in degrees of each exterior angle of several regular convex polygons. If the pattern in the table continues, which statement is true?
| Number of Sides | Exterior Angle Measure (degrees) |
|---|---|
| 3 | 120 |
| 4 | 90 |
| 5 | 72 |
| 6 | 60 |
| 10 | 36 |
| 15 | 24 |
| 20 | 18 |
| 45 | 8 |
| 90 | 4 |
- As the number of sides in the polygon increases by 1, the measure of each exterior angle decreases by 25%.
- The product of the number of sides in the polygon and the measure of each exterior angle is a constant.
- The measure of each exterior angle is a multiple of 6 degrees.
- If the number of sides in the polygon is even, the measure of each exterior angle is a multiple of 3 degrees.
Show STAAR answer and explanation
The product of the number of sides in the polygon and the measure of each exterior angle is a constant.
The exterior angles sum to 360°, so for a regular n-gon each angle is . Their product is always .
As the number of sides in the polygon increases by 1, the measure of each exterior angle decreases by 25%.: The change from 4 to 5 sides reduces the angle from 90° to 72°, a 20% decrease, not 25%.
The measure of each exterior angle is a multiple of 6 degrees.: The table includes 8° and 4°, neither a multiple of 6°.
If the number of sides in the polygon is even, the measure of each exterior angle is a multiple of 3 degrees.: A regular 90-gon has 4° exterior angles, not multiples of 3°.
Learning objectives
By the end of the lesson, students can:
- Explain why one exterior angle at each vertex of a convex polygon totals 360°.
- Find the exterior angle or number of sides of a regular polygon.
- Identify the constant product linking side count and exterior-angle measure.
Instructional sequence
- 1
Engage
5 minWalk around a taped polygon and turn at each corner. Ask how much the walker has turned upon facing the starting direction again.
- 2
Investigation
12 minMeasure exterior angles of several convex polygons and add them. Compare regular and irregular examples.
- 3
Direct instruction
10 minGeneralize the 360° total and derive n times e equals 360 for regular polygons.
- 4
Guided practice
12 minComplete tables, find missing side counts, and test claims about fixed differences or fixed percentage decreases.
- 5
Closure
6 minComplete the exit ticket and identify the constant quantity in the table.
Misconceptions to address
Teacher move: The total is 360° for every convex polygon, but equal individual angles require an equiangular polygon, including every regular polygon.
Teacher move: Compare successive values of 360/n. Their differences change; the constant is the product n times the exterior-angle measure.