Angles,Polygons And Geometric Formulas

Angles:

Types of angle	How it is
Acute angle	Angle less than 900
Right angle	At 900
Obtuse angle	Angle greater than 900  and less than1800
Straight angle	Angle exactly 1800
Reflex Angle	Angle greater than 1800

 Supplementary Angle : when two angles added up to 180⁰

Complementary Angle : when two angles added up to 90⁰

When a traversal intersects with two parallel lines eight angles are produced.

Corresponding angles are equal Example:

                                          1 and 5 angle are equal and 4 and 8 are also equal.

Alternate interior angles are equal . Example :

                                          2 and 8 are equal and 3 and 5 are equal. 

Alternate Exterior angle are equal . Example:

                                           4 and 6 are equal and 1 and 7 are equal.

Consecutive interior angles are equal to 180 . Examples :

                                 2 + 5 are equal to 180⁰ and 3 + 8 are equal to the 180^0.

Vertical Angles : two opposite angles are equal . Example :

                                         5 and 7 are equal and 6 and 8 angle are equal .

                                        1 and 2 and 1 and 4 are also supplementary angles.

Polygons: simple closed curve made up of only line segments is called Polygon.

Diagonal is the line segment connecting two non consecutive vertices of a polygon.

Angle sum properties:

Polygon: A closed plane figure made up of several line segments that are joined together. The sides do not cross each other. Exactly two sides meet at every vertex.

Types of Polygons
Regular - all angles are equal and all sides are the same length. Regular polygons are both equiangular and equilateral.
Equiangular - all angles are equal.
Equilateral - all sides are the same length.

	Convex - a straight line drawn through a convex polygon crosses at most two sides. Every interior angle is less than 180°.
	Concave - you can draw at least one straight line through a concave polygon that crosses more than two sides. At least one interior angle is more than 180°.

Polygon Formulas:

(N = # of sides and S = length from center to a corner)

The number of diagonals in a polygon = 1/2 N(N-3)

The number of triangles (when you draw all the diagonals from one vertex) in a polygon = (N - 2)

Area of a regular polygon = (1/2) N sin(360°/N) S²

Sum of the interior angles of a polygon = (N - 2) x 180°

Polygon parts:

Side - one of the line segments that make up the polygon.

Vertex - point where two sides meet. Two or more of these points are called vertices.

Diagonal - a line connecting two vertices that isn't a side.

Interior Angles - Angle formed by two adjacent sides inside the polygon.

Exterior Angles - Angle formed by two adjacent sides outside the polygon.

Polygon	No of side	No of Angles	No of vertices	No of Diagonals
Triangle	3	3	3	0
Quadrilateral	4	4	4	2
Pentagon	5	5	5	5
Hexagon	6	6	6	9
Heptagon	7	7	7	14
Octagon	8	8	8	20
Nonagon	9	9	9	27
Decagon	10	10	10	35

Sum of the Exterior angles = 360⁰

Sum of the Interior angles = 180⁰* (n-2)

                                               N is number of sides.

How to find out number of sides of a polygon:

If we know one exterior angle of regular polygon

                                       Number sides   = 360⁰ / angle of give

How to find out exterior angles :

If we know number of sides

         One of the  Exterior angle =360⁰/12

How to find out exterior angle if we know interior angle:

Exterior angle + interior angle = 180⁰( because they are linear pairs)

                                             

                                          Quadrilaterals

Quadrilateral just means "four sides"
(quad means four, lateral means side).

A Quadrilateral has four-sides, it is 2-dimensional (a flat shape), closed (the lines join up), and has straight sides.

Four sides, four vertices, sum of the all interior angles are 360.

 For example a square, rhombus and rectangle, kite ,trapezium are also parallelograms.

Quadrilaterials are two types

Parallelograms  and  non parallelograms

Parallelograms - square, rhombus, parallelogram, rectangle, and the trapezoid.

Non Parallelograms : Trapezium and Kite

Perimeter formula
Square	4 × side
Rectangle(kite)	2 × (length + width)
Parallelogram	2 × (side1 + side2)
Triangle	side1 + side2 + side3
Regular n-polygon	n × side
Trapezoid	height × (base1 + base2) / 2
Trapezoid	base1 + base2 + height × [csc(theta1) + csc(theta2)]
Circle	2 × pi × radius
	Area formula Rectangle length × width Parallelogram base × height Triangle base × height / 2 Regular n-polygon (1/4) × n × side2 × cot(pi/n) Trapezoid height × (base1 + base2) / 2 Circle pi × radius2 Kite Pq/2 Cube (surface) 6 × side2 Sphere (surface) 4 × pi × radius2 Cylinder (surface of side) perimeter of circle × height 2 × pi × radius × height Cylinder (whole urface) Areas of top and bottom circles + Area of the side 2(pi × radius2) + 2 × pi × radius × height Cone (surface) pi × radius × side Torus (surface) pi2 × (radius22 - radius12) Volume formula Cube side3 Rectangular Prism side1 × side2 × side3 Sphere (4/3) × pi × radius3 Ellipsoid (4/3) × pi × radius1 × radius2 × radius3 Cylinder pi × radius2 × height Cone (1/3) × pi × radius2 × height Pyramid (1/3) × (base area) × height Torus (1/4) × pi2 × (r1 + r2) × (r1 - r2)2.