Polygons — Interior and Exterior Angles

ACT Math · Axiom Academy

LESSON Polygons: Interior & Exterior Angles Three formulas cover every ACT polygon-angle question — and each one is just a picture: triangles fanning out, angles closing a circle. 1. The Three Angle Formulas You Need Here n is the number of sides. The whole reason the first formula works: from a single vertex you can slice any polygon into (n-2) triangles , and each triangle carries . Watch the fan of triangles grow as n climbs from a triangle to a hexagon. 2. Angle Reference Table — Memorize the Common Ones The ACT most often tests pentagons, hexagons, and octagons . Having these values ready saves real time. The animation cycles a regular polygon from n=3 up to n=8 , showing each interior angle opening wider as n grows. 3. Exterior Angles: The 360 Rule The exterior angles of any convex polygon — regular or not — always sum to exactly . Here's why: walk once around the boundary and at each corner you turn by that vertex's exterior angle. By the time you return to the start, facing your original direction, you've turned through one full circle. Watch the five turns of a pentagon stack up into . Worked Example: Finding n from an Exterior Angle Each exterior angle of a regular polygon measures . How many sides does the polygon have? Now run the exterior-angle logic in reverse. If each turn is , a regular polygon closes up after exactly turns — so it has 15 sides. Watch the polygon build itself edge by edge, each vertex adding one turn until it snaps shut.

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