POLYGONS

We studied about triangle and quadrilaterals.

A triangle has three sides and three angles.

A quadrilateral has four sides and four angles.Let us study about the shapes with more sides.If a shape has 5 sides and 5 angles then it is known as pentagon.If a shape has 6 sides and 6 angles it is known as hexagon.If a shape has 7 sides and 7 angles is known as heptagon.Similarly octagon, nonagon , decagon.There is a common name for all these together polygonsA polygon is a simple closed curve formed of only line segments . A triangle is a very common example.CONVEX AND CONCAVE POLYGONS

A polygon is said to be convex if it has no portion of its exterior otherwise it is said to be a concave polygon

A quadrilateral can be split in to two triangles by drawing a diagonal.Similarly in a pentagon ,if we choose a vertex ,skip the one near it and join with the next vertex ,we can divide it in to a quadrilateral and a triangle. In a hexagon if we choose a vertex skip the one near it and join with the next vertex. We can divide it in to a pentagon and a triangle.Thus we can divide any polygon in to another polygon with one side less than a polygon and a triangle by drawing a line starting at any vertex and joining with the next.TRIANGULAR DIVISION`POLYGONNO OF SIDESNO:OF TRIANGLE

QUADRILATERAL42 (4-2=2)

PENTAGON53(5-2=3)

HEXAGON64(6-2=4)

HEPTAGON75(7-2=5)

OCTAGON86(8-2=6)

In general , what is the relation between the number of sides of the polygons and the number of triangle we get like this? The number of triangles=number of sides of polygon-2That is the number of triangles got by dividing polygon is two less than the number of sides.SUM OF THE ANGLES

We know that the sum of the angles of a triangle is 180 and the sum of the angles of a quadrilateral is 360. How we found that the sum of the angles of a quadrilateral is 360?

When we draw a diagonal of a quadrilateral, it is split in to two triangles. The diagonal also divides in to two , the angle at each of the vertex it passess through one part in one triangle and the other part in the other triangle.Thus the angles of the quadrilateral become the angles of these two triangles, and so their sum is 2 X 180=360Similarly we can compute the sum of the angles of a pentagon,it is split in to a quadrilateral and triangle.

Sum of the angles of a pentagon=Sum of the angles of a quadrilateral+Sum of the angles of the triangle=360+180=540Can you find like this the sum of the angles of hexagon by splitting it into a pentagon and a triangle . Sum of the angles of a hexagon=sum of the angles of the pentagon +sum of the angles of the triangle=540+180=720Generally speaking we can divide a polygon in to a previous polygon and a triangle; and so its sum of the angles will be 180more the sum of the angles of the previous one.So can you say what the sum of the angles of a ten sided polygon?How about a twenty sided polygon?20 sides means , 17 sides more than a triangle , and this means the sum increases by 17 x 180The sum of the angles of a 20 sided polygon=180+ (17 x 180) = 18 x 180 =3240 We can write this as a general principle , using a bit of algebra. To get an n sided polygon,we start with a triangle and add n-3 sides more. Adding each side increases the sum of angles by 180.So the sum of the angles of an n sided polygon is (n-3) x 180 more than the sum of the angles of a triangle. Thatis the sum of the angles is =180 + (n-3) x180=1+(n-3) X180=(n-2) x 180The sum of the angles of an n sided polygon is (n-2) x 180

EXTERNAL ANGLES

In triangle ABC , the side BC is extended and this produces a new angle outside the triangle. This angle ACD is called an external angle of the triangle.What is the relation between this angle and the angle ACB of the triangle?