Polygons and their angles
A polygon is a flat, closed shape made of straight sides. A polygon with n sides also has n corners, and at each corner the two sides make an interior angle, the angle inside the shape. A pentagon has 5 sides and 5 interior angles; a hexagon has 6 of each.
The interior angles of a triangle add up to 180°. The same fact gives the total for any polygon, by cutting the polygon into triangles.
Cut it into triangles from one corner
Take a pentagon and choose one corner. From that corner, draw a straight line to every corner it is not already joined to by a side. A line joining two corners that are not next to each other is called a diagonal. The two diagonals from one corner of a pentagon cut it into three triangles.
Every corner of every triangle sits at a corner of the pentagon, and at each corner of the pentagon the triangles’ angles fit side by side and fill the pentagon’s angle exactly. So the angles of the three triangles, taken together, are exactly the interior angles of the pentagon, nothing more and nothing less.
Each triangle’s angles add up to 180°, so the pentagon’s interior angles add up to 3 × 180 = 540°.
Two diagonals from the top corner cut the pentagon into three triangles. Their angles together make the pentagon’s angles: 3 × 180° = 540°.
How many triangles
A hexagon cut the same way makes four triangles, so its interior angles add up to 4 × 180 = 720°.
The count always comes out two less than the number of sides. Each triangle in the fan stands on one side of the polygon, the side opposite the chosen corner, and no two triangles share that side. The two sides that meet at the chosen corner are the only sides with no triangle standing on them. So an n-sided polygon makes n − 2 triangles, and its interior angles add up to (n − 2) × 180°.
A hexagon cut from one corner makes four triangles: 4 × 180° = 720°.
The totals
A triangle is 1 × 180 = 180°. A quadrilateral is 2 × 180 = 360°. A pentagon is 3 × 180 = 540°, a hexagon 4 × 180 = 720°, and an octagon, with 8 sides, 6 × 180 = 1080°. Each extra side adds one more triangle, so one more 180°.
The total finds a missing angle. A quadrilateral with angles of 80°, 95° and 110° has a fourth angle of 360 − 80 − 95 − 110 = 75°.
The usual mistakes
Counting a triangle for every side, n × 180. Cutting from a point in the middle of the shape does make n triangles, but their angles around that middle point add up to a full turn, 360°, and those are not angles of the polygon. Take them away: n × 180 − 360 = (n − 2) × 180, the same answer.
Using 360° for every polygon. 360° is the total for a quadrilateral only; a pentagon’s angles add up to 540°.
Counting one triangle too many, (n − 1) × 180. Check with a quadrilateral: one diagonal cuts it into 2 triangles, two fewer than its 4 sides.
A polygon with a dent
A polygon can have a dent, where one interior angle is bigger than 180°, a reflex angle. The total is still (n − 2) × 180°: cut it into triangles from the corner of the dent. The reflex angle counts at its full size, the angle inside the shape.
Worked example: Angle Sum of a Polygon
Question A hexagon has angles of 130°, 95°, 140°, 110° and 125°. Find its sixth angle.
1.Diagonals from one vertex cut a hexagon into 6 − 2 = 4 triangles.
From one vertex, three diagonals cut the hexagon into four triangles. 2.Angle sum = 4 × 180° = 720°.
4 × 180° = 720° for the six angles together. 3.The five known angles: 130 + 95 + 140 + 110 + 125 = 600.
The five given angles add to 600°. 4.Sixth angle = 720° − 600° = 120°.
The sixth: 720° − 600° = 120°.
Answer: 120°
Common mistakes
- Using 360°, the angle sum of a quadrilateral, for every polygon.
- Counting 6 triangles instead of 4: the two vertices next to the starting one make no triangle of their own.
Worked example: Irregular Polygon with a Reflex Angle
Question A pentagon has angles of 95°, 110°, 70° and a reflex angle of 230°. Find its fifth angle.
1.A pentagon splits into 5 − 2 = 3 triangles, so its angles add to 3 × 180° = 540°.
Three triangles from the dented vertex: 3 × 180° = 540°. 2.The reflex angle counts as 230°, the inside of the dent.
The reflex angle is the 230° inside the dent. 3.The four known angles: 95 + 110 + 70 + 230 = 505.
95 + 110 + 70 + 230 = 505. 4.Fifth angle = 540° − 505° = 35°.
The fifth angle: 540° − 505° = 35°.
Answer: 35°
Common mistakes
- Replacing the reflex angle by 360° − 230° = 130°, the angle outside the dent, which is not an angle of the polygon.
- Refusing the answer because 35° looks small next to 230°: a sharp point opposite a dent is exactly what a reflex angle makes.