Angles in Polygons A polygon is a CLOSED shape where each edge is a straight line and all edges meet at vertices. A polygon with 3 sides is called a triangle. A polygon with 4 sides is called a quadrilateral. Squares and rectangles are examples of quadrilaterals. The names of polygons is based on how many sides they have. Here is a list of some of them: 3 sides: Triangle 4 sides: Quadrilaterals 5 sides: Pentagon 6 sides: Hexagon 7 sides: Heptagon 8 sides: Octagon 9 sides: Nonagon 10 sides: Decagon Here is an extensive list from Wikipedia: Not all polygons have special names. Some, like a 51-­‐sided shape, is simply called a 51-­‐gon. And 87-­‐sides would be called an 87-­‐gon. Sum of Interior Angles in a Triangle The sum of the 3 angles in a triangle is 180. This can easily be shown by using 2 transversals meeting at a point on one of the parallel lines to make a triangle. a + b + c = 180 because, using alternate angles, it can be seen that ‘b’ is the same as the angle adjacent to ‘a’ on the left. And also using alternate angles, it can be shown that ‘c’ is the same as the angle adjacent to ‘a’ on the right. Now it should be obvious, that a + b + c = 180 because of the straight line. Sum of Interior Angles in a Quadrilateral The sum of the 4 interior angles in a quadrilateral is 360. This can be shown because any quadrilateral can be cut into 2 triangles. Note the quadrilateral below, its 4 interior angles are: a, (b+f), e, and (d+c). Note 2 angles have been cut into 2 to make 6 angles. Now a + b + c = 180 (it’s a triangle). And d + e + f = 180 (its also a triangle). So the 6 angles (which are equivalent to the 4 interior angles of the quadrilateral) add up to 2 lots of 180 = 360. Sum of Interior Angles for any polygon Each polygon has its own “rule”, just like triangles and quadrilaterals. The rule can be worked out using a formula. With a quadrilateral, we made 2 triangles. With a pentagon, we can start at a vertex and cut it up into 3 triangles, and the sum of the interior angles of the pentagon coincide with the sum of the interior angles of the 3 triangles. And 3 x 180 = 540. So the “rule” for a pentagon is the sum of the interior angles is 540 degrees. With a hexagon, we can cut it into 4 triangles. So the rule is 4 lots of 180 which is 720 degrees = the sum of the interior angles. In general, for any polygon, you can pick a vertex and make triangles by joining that vertex up to EVERY other vertex which isn’t adjacent to it (if a vertex is adjacent, they are already joined!!!). By drawing a couple it is easy to see that the number of triangles made is equal to the number of sides – 2. So quadrilaterls make 4 -­‐2 = 2 triangles. Pentagons make 5 -­‐2 = 3 triangles. Hexagons make 6 – 2 = 4 triangles. And… since we know each triangle has interior angles adding up to 180, the sum of the interior angles of the polygon is just 180 multiplied by (the number of sides – 2). RULE: Angle Sum = (number of sides – 2) x 180. e.g. For a decagon (10 sides), the sum of the angles will be (10 – 2) x 180 = 8 x 180 = 1440. An alternative way of getting the SAME result is to cut the polygon differently. This way requires you to put a dot in the middle, and then join all the vertices to this dot. This will create a number of triangles equal to the number of sides. Now we know the interior angles in each triangle is 180. Notice that the polygon’s exterior angles is equal to the triangles interior angles – 360 (in the middle) which aren’t part of the exterior of the polygon. So we can say Sum of Interior Angles = no. of sides x 180 – 360. By expanding (no. of sides – 2) x 180, you actually get no. of sides x 180 – 360.