Tenth Grade · Math · week 30

Polygons: interior/exterior angle sums and regular polygons

Compute the sum of a polygon's interior angles, and find interior or exterior angle measures of a regular polygon in either direction.

Any polygon's interior angles can be found by chopping it into triangles from one vertex: a polygon with n sides splits into (n-2) triangles, and since each triangle contributes 180 degrees, the SUM of a polygon's interior angles is (n-2) * 180 degrees. A quadrilateral (n=4) splits into 2 triangles, giving 360 degrees; a pentagon (n=5) splits into 3 triangles, giving 540 degrees, and so on.

Exterior angles behave more simply: no matter how many sides a polygon has, its exterior angles (one per vertex, turning the same direction all the way around) always add up to exactly 360 degrees -- one full turn walking the perimeter. For a REGULAR polygon (all sides congruent, all angles congruent), dividing evenly gives a formula for a single angle: each interior angle is (n-2)*180 / n, and each exterior angle is 360 / n. Since an interior angle and its exterior angle form a linear pair, checking that the two you compute add to 180 is a fast built-in error check.

📋 For the teacher
Teach it (10–15 min):
  • Have them draw the diagonals from ONE vertex of several polygons (quadrilateral, pentagon, hexagon) and count the triangles formed, so (n-2) emerges from counting rather than a memorized formula.
  • Walk an actual pentagon or hexagon shape (masking tape on the floor) and note the total turning is exactly one full rotation no matter the shape -- that is the exterior angle sum, felt physically.
  • Practice the "solve for n" direction explicitly (given an interior or exterior angle, find the number of sides), since it runs the formula backward and trips people up more.
  • Use the interior-plus-exterior-equals-180 check as a standard habit on every regular polygon problem.
Talk about it:
  • Why does splitting a polygon into triangles from one vertex always give exactly (n-2) triangles?
  • Why do a polygon's exterior angles always add to 360 degrees, no matter how many sides it has?
  • As a regular polygon gets more sides, what happens to each interior angle? Each exterior angle?

Mastery looks like: They compute the sum of interior angles, and the measure of each interior or exterior angle of a regular polygon, in both directions.

Common stumbles: Using n instead of (n-2) in the interior angle sum formula; forgetting the exterior angle formula always uses 360, not the interior angle sum.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

AdditionSee groups combine — with counters, blocks, and the number line.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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