ACT Math: Interior and Exterior Angles of Polygons—Formulas and Applications

Interior angles: The angles inside a polygon. Sum of interior angles=(n-2)×180°, where n=number of sides. Example: Triangle (n=3): (3-2)×180°=180°. Quadrilateral (n=4): (4-2)×180°=360°. Pentagon (n=5): (5-2)×180°=540°. For a regular polygon (all angles equal), each interior angle=[(n-2)×180°]/n. Example: Regular hexagon: [(6-2)×180°]/6=720°/6=120°. Exterior angles: The angles outside a polygon (supplementary to interior angles). Sum of exterior angles=360° for any polygon. Each exterior angle of a regular polygon=360°/n. Example: Regular hexagon: 360°/6=60°. Note: Interior+exterior=180° at each vertex. On the ACT, you'll use these formulas to: (1) Find unknown angles in polygons, (2) Determine the number of sides given angle measures, (3) Work with regular polygons. The sum of exterior angles is always 360°, regardless of the number of sides—this is a powerful check for problems.

Why exterior angles matter: They're often simpler to work with; they always sum to 360°, which is easier than the (n-2)×180° formula.

Mistake 1: Forgetting (n-2) in the interior angle formula. Sum is NOT n×180°; it's (n-2)×180°. Mistake 2: Confusing interior and exterior. Interior angles are inside; exterior are outside (supplementary). Mistake 3: Assuming all polygons are regular (angles equal). Many are irregular (angles vary). Mistake 4: Misapplying the 360° exterior angle rule. It's the sum, not each angle. Each exterior angle of a regular pentagon is 360°/5=72°, not 72° for all irregular pentagons. Always verify: Interior+exterior at a vertex=180°. Sum of all exterior angles=360°.

Checklist: (1) Identify n (number of sides). (2) Determine if regular (all angles equal) or irregular. (3) Use correct formula. (4) If finding unknown angles, set up an equation and solve. (5) Verify the sum.

Problem 1: Sum of interior angles of a heptagon (7-sided)? (7-2)×180°=900°. Problem 2: Each interior angle of a regular octagon? [(8-2)×180°]/8=1080°/8=135°. Problem 3: Each exterior angle of a regular decagon? 360°/10=36°. Problem 4: A polygon has interior angle sum of 1440°. How many sides? (n-2)×180°=1440°. n-2=8. n=10 (decagon). Problem 5: An irregular hexagon has five angles: 100°, 120°, 130°, 110°, 125°. Find the sixth. Sum must be 720°. Sixth angle=720°-585°=135°. Problem 6: If each exterior angle of a regular polygon is 20°, how many sides? 360°/20°=18 sides. For each, show the formula you used and verify your answer makes sense (interior+exterior=180° at each vertex).

Daily drill: Solve one polygon problem daily. Alternate between interior, exterior, and finding missing angles. Practice both regular and irregular polygons.

About 1-2 geometry questions per ACT Math section involve polygon angles. These are usually straightforward formula applications—earn points by knowing the formula and applying it correctly. Many students forget the (n-2) factor or confuse interior and exterior, making correct answers less common and more valuable for your relative score.

Spend 1-2 days on polygon angles. Memorize the formulas, drill the relationship between interior and exterior, and practice problems. By test day, you'll solve polygon angle questions quickly and correctly.