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Interior Angles of Polygons An Interior Angle is an angle inside a shape. Triangles The Interior Angles of a Triangle add up to 180° 90° + 60° + 30° = 180° It works for this triangle! 80° + 70° + 30° = 180° Let's tilt a line by 10° ... It still works, because one angle went up by 10°, but the other went down by 10° Quadrilaterals (Squares, etc) (A Quadrilateral is any shape with 4 sides) 90° + 90° + 90° + 90° = 360° A Square adds up to 360° 80° + 100° + 90° + 90° = 360° Let's tilt a line by 10° ... still adds up to 360°! The Interior Angles of a Quadrilateral add up to 360° Because there are Two Triangles in a Square The internal angles in this triangle add up to 180° (90°+45°+45°=180°) ... and for this square they add up to 360° ... because the square can be made from two triangles! Pentagon A pentagon has 5 sides, and can be made from three triangles, so you know what ... ... its internal angles add up to 3 × 180° = 540° And if it is a regular pentagon (all angles the same), then each angle is 540° / 5 = 108° (Exercise: make sure each triangle here adds up to 180°, and check that the pentagon's internal angles add up to 540°) The General Rule So, each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total: If it is a Regular Polygon.. Shape Sides Sum of Internal Angles Triangle 3 180° 60° Quadrilateral 4 360° 90° Pentagon 5 540° 108° Hexagon 6 720° 120° ... ... .. Any Polygon n (n-2) × 180° Shape Each Angle ... ... (n-2) × 180° / n That last line can be a bit hard to understand, so let's have one example. Example: What about a Decagon (10 sides) ? Sum of Internal Angles = (n-2) × 180° = (10-2)×180° = 8×180° = 1440° And, if it is a Regular Decagon, then each internal angle = 1440°/10 = 144°