Kōkiri Learn

Background reading · Mathematics and Statistics

Angles, triangles and the 180° rule

Types of triangles, why their angles always add to 180°, and how that rule unlocks the angle sum of any polygon.

Triangles are the strongest and simplest flat shape. Bridges, roof trusses and power pylons are full of them. They also hold a secret that unlocks every other polygon.

Naming triangles

You can name a triangle by its angles or by its sides.

  • Acute triangle: all three angles are less than 90°. - Right triangle: one angle is exactly 90°. - Obtuse triangle: one angle is more than 90°. - Equilateral: three equal sides and three equal angles of 60°. - Isosceles: at least two equal sides, and the two angles opposite them (the base angles) are equal. - Scalene: all sides are different lengths.

    • The 180° rule

    Tear the three corners off any paper triangle and put them side by side. They always make a straight line, which is 180°. When you measure with a protractor you might get 178° or 183°, because nobody measures perfectly. Mathematicians prove the rule is always exactly 180° by drawing a line through one corner parallel to the opposite side and matching up equal angles.

    From triangles to any polygon

    A polygon is a flat shape with straight sides. Pick one corner and draw lines to the other corners. A quadrilateral splits into 2 triangles, so its angles add to 2 × 180° = 360°. A pentagon splits into 3 triangles (540°) and a hexagon into 4 (720°). There are always two fewer triangles than sides, so the rule is:

  • Angle sum of a polygon: 180 × (n − 2) degrees, where n is the number of sides. - Regular polygon: all sides and angles are equal, so each angle is the angle sum divided by n. - Exterior angle: the turn you make at each corner as you walk around the outside. For every polygon, the exterior angles add to 360°, one full turn.

    • Why it matters for tiles

    When tiles meet at a point, their angles must add to exactly 360° or there will be a gap or an overlap. Knowing the angles of your shapes lets you predict whether they will fit before you cut a single tile.

    Sources and further reading

    Written for Kōkiri Learn students in our own words. Check facts against the sources.

    Used in: Shape Shifters: Transformations and Tessellation