Polygon questions on the ACT usually ask for a missing interior angle, the total angle sum, or a side length in a regular figure—using (n − 2) × 180° for n sides and, when the polygon is regular, dividing that sum evenly. Triangles are the most common polygon; everything else builds on the same angle-sum rule.
The interior angle sum formula
Any n-sided polygon has interior angles that add to (n − 2) × 180°. A triangle (n = 3) gives 180°, a quadrilateral 360°, a pentagon 540°, a hexagon 720°. If five angles of a hexagon are 120°, 130°, 110°, 125°, and 115°, the sixth is 720° minus the sum of those five, or 120°.
Regular polygons
In a regular polygon, every side and every interior angle is equal. Each interior angle is (n − 2) × 180° divided by n. For a regular octagon (n = 8), each interior angle is 6 × 180° ÷ 8 = 135°. The ACT may also ask for exterior angles—each is 360° ÷ n for a regular n-gon, and exterior angles always sum to 360° even when the polygon is not regular.
Diagonals (occasionally tested)
From one vertex of an n-gon, you can draw n − 3 diagonals. Total diagonals in the whole figure: n(n − 3) ÷ 2. A hexagon has 9 diagonals. This shows up less often than angle sums but is quick points if you know the formula.
Triangles are still the workhorse
Most "polygon" setups reduce to triangles once you draw a diagonal or an extra segment. If you are shaky on triangle angle sums and area, start with ACT triangle problemsbefore drilling higher n-gons.
Perimeter and area of regular figures
Perimeter of a regular n-gon is n times side length. Area sometimes uses one half times apothem times perimeter, but the ACT more often splits the figure into triangles and rectangles you already know. See area and volume formulas for the flat shapes that appear inside composite figures.
Where polygons sit in ACT geometry
Polygons share the geometry pillar with lines, circles, and 3D figures. For the full topic map, see how much geometry is on the ACT and lines and angles on ACT Mathwhen parallel sides of a polygon are involved.
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