Basic Geometry

Introduction to Geometry

Geometry is a branch of mathematics concerned with problems finding the areas of plane figures such as triangles, rectangles, circles, etc.

Image Src: Google
And if you are facing a problem in geometry then it will be very helpful to draw yourself a diagram.

Different Terms in Geometry:
There are some basic concepts in geometry, namely, "points", "line", "plane" and "solid shapes". It is not possible to precise these concepts precisely. We can use a small dot to make a figure of a point. When two lines intersect we get a point.

Dimensions of geometric objects. 
Point - No Dimensions.
Line - One Dimension.
Plane - Two Dimensions.
Solid - Three Dimensions.

Points

A point can be shown as a fine dot made by a sharp pencil on a paper. A point cannot be described in terms of length, breadth, and height, because it is unitless as well as dimensionless. However, in geometry, everything starts with a point if you have to draw a line, a plane even a three-dimensional geometry.

Lines:

A line is a collection of points and extends endlessly in both directions. To emphasize this point we use arrowheads in both the directions. If two lines are intersecting that means they have a common point which is known as the "point of intersection".
                                   
If there is a line that contains three or more points are said to be Collinear Points.
If there is a point which lies on three or more lines are said to be Collinear Lines.

Line segments and rays

The line segment starts at a specific point, and go to another, the endpoint. They are drawn as a line between two points, as you would probably expect the figure given below.
Ray is the second type of line and it will go on forever. They are often drawn as a line starting from a point with an arrow on the other end, you can see in a figure given below:


                                           

Parallel and perpendicular lines

Parallel lines never meet or intersect with each other or the distance between both lines is always constant.
Perpendicular lines always intersect at a right angle, 90°:
                            
Note: Many line segments may be joined to form a different geometric figure.

Two Dimensional shapes and their properties:

Knowing the properties of 2-D shapes is a basic but important skill. Depending on their shapes their properties may vary. Under which we have explained the properties of such different shapes.

Circle

A circle is formed by tracing a point that moves in a plane such that its distance from a fixed point(center) is constant and the picture of that is shown below:
                                                 
𑂽 The fixed point in the circle is called center.
𑂽 Radius is the fixed distance between the center and set of points. It is denoted by "R".
     Hence, Diameter = Twice the length of the radius or D=2*R.
𑂽 A chord is a line segment whose endpoint lies on the circle.
𑂽 A line that touches the circle at any point is known as a tangent.
The radius of the circle is always perpendicular to the tangent at the point where it touches the circle.

Properties:

  • Circumference of the circle = 2 × π × Radius.
  • Length of an arc= (Central angle made by the arc/360°) × 2 × π × Radius.
  • Area of the circle = π × (Radius)².
  • Area of the circular sector =(Central angle made by the sector/360°) × π × (Radius)².

Triangle

A triangle is a shape bounded by three lines in a plane. More precisely, you can say that it is a polygon. There are three sides and three angles may be equal or different. The sum of all the angles of a triangle is 180°.
Triangles are classified into various types on the basis of the lengths of their sides as well as on the basis of the measures of their angles.
                                                     
Following are the types based on their sides:

Scalene Triangle: A triangle, no two of whose sides are equal.
Isosceles Triangle: A triangle, two of whose sides are equal in length.
Equilateral Triangle: A  triangle, all of whose sides are equal.

Following are the types of the triangle based on their angles:

Acute Triangle: A triangle each of whose angle measures is acute (less than 90∘).
Right Triangle: A triangle with one angle at a right angle (=90∘).
Obtuse Triangle: A triangle with one angle an obtuse angle (more than 90∘ but less than 180∘).
Note: It should be noted that an equilateral triangle is an isosceles triangle but the converse is not true

Properties:

  • A triangle can not have more than one right angle.
  • In a right triangle, the sum of two acute triangles is 90°.
  • A triangle can not have more than one obtuse angle.
  • Angles opposite to two equal sides of a triangle are equal.
  • The addition of two sides of the triangle is always greater than the third side.
  • The difference between the two sides of a triangle is less than the third side.
  • Exterior angle property: The sum of the interior opposite angle of a triangle is always equal to its corresponding exterior angle.
  • If corresponding angles of two triangles are congruent and the length of their sides are proportional then it is said to be a similar triangle.

Some basic formulas:
Area=1/2 ✖ Base ✖ Height
Perimeter=Addition of the length of all three sides.
By Pythagorean theorem , there is a relation between base, hypotenuse, and perpendicular sides of a right triangle.
                    (Hypotenuse)² = (Base)² + (Perpendicular)².

Quadrilateral:

If we have four points in a plane such that no three of them are collinear, then the figure obtained by joining them in pairs is called a quadrilateral. the sum of all four angles of the quadrilateral is 360°.
Various types of Quadrilaterals are:

Trapezium
A quadrilateral having one pair of parallel sides is called a trapezium. Two pairs of adjacent angles add up to 180°.
                                    
Properties:

  • It is said to be an Isosceles trapezium if its non-parallel sides are equal.
  • The line joining the midpoints of the non-parallel sides is parallel to the parallel sides and is equal to half the sum of the parallel sides.
  • One pair of adjacent angles can be a right angle. The other pair of adjacent angles will be supplementary angles but not right angles.

Parallelogram

A parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides and angles of a parallelogram are of equal length and equal measure respectively.
                                         
Properties:

  • Area: base × height
  • Perimeter: 2 x (sum of lengths of adjacent sides)
  • Number of vertices: 4
  • Number of edges: 4
  • Type: Quadrilateral
  • Line of symmetry: 0

Line of Symmetry:  An axis or an imaginary line that passes through the center of the object or shape and slices up it into identical halves, so that the sliced part came out mirror images of each other i.e called the axis of symmetry.

Rhombus

A rhombus is a quadrilateral whose four sides all have an equal length. Another name of the rhombus is an equilateral quadrilateral, since equilateral means that all of its sides have the same length. It also has the shape of a kite.
                                           
Properties:

  • Area: ½ x (product of the lengths of the diagonals)
  • Perimeter: 4 x side
  • Number of vertices: 4
  • Number of edges: 4
  • Line of symmetry: 2

Rectangle

A rectangle is a quadrilateral with four right angles. It can also be defined as a quadrilateral with all angles are equal, it means that all of its angles are equal. It can also be defined as a parallelogram containing right angles.
                                             
Properties:


  • Area: length×width
  • Perimeter: 2 x (length+width)
  • Number of vertices: 4
  • Number of edges: 4
  • Internal angle: 90°
  • Line of symmetry: 2, Lines going through the midpoint of opposite sides.

Square

A square is a regular quadrilateral, which means that it has four congruent sides and interior angles measure 90 degrees. It can also be defined as a rectangle in which two adjacent sides have equal length.
                                         
Properties:

  • Area: (side)²
  • Perimeter: 4 x side
  • Number of vertices: 4
  • Number of edges: 4
  • Internal angle: 90°
  • Line of symmetry: 4, Two diagonals, and two lines going through the midpoint of opposite sides.

Three Dimensional shapes and their properties:

cone

A cone is a three-dimensional geometric shape that becomes narrower smoothly from a circular bottom to a point called peak or vertex. A cone is formed by a set of line segments, linking a common point, the vertex, to all of the points on a circular base that is in a plane that does not contain the vertex. A cone with a polygonal base is called a pyramid.
The axis of a cone is the straight line (if any), passing through the apex, about which the base (and the whole cone) has a circular symmetry.
Properties:

  • Volume: π × r² × (h/3)
  • surface area: πrl + πr²
  • Number of faces: 1
  • Base shape: Circle     

Cylinder

A cylinder has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. The surface is formed by tracing the point at a fixed distance from a given line segment called the axis of the cylinder.
An oblique cylinder  has both bases not aligned above each other and the axis of cylinder not at 90 degrees to both circular faces.
Properties:

  • Volume: π × r² × h
  • surface area: 2πrh+2πr²
  • Number of faces: 2
  • Base shape: Circle
  • Top shape: Circle
  • Properties of right circular cylinder:

Cube

A cube is a three-dimensional solid object bounded by six square faces, with three meetings at each vertex. It has 6 faces, 12 edges, and 8 vertices.
                                             
Properties:

  • Volume: a³
  • Number of edges: 12
  • Number of vertices: 8
  • Number of faces: 6
  • Side shape: Square Base shape: Square

Cuboid

A cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. more closely cuboid is made up of 6 rectangles that are placed at right angles with each other.
                                     
Properties:

  • Volume: l × b × h
  • Number of edges: 12
  • Number of faces: 6
  • Number of vertices: 8

Tetrahedron

A tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".                                         

Properties:  

  • Volume: (√2)/12 × a³
  • Number of faces: 4
  • Number of edges: 6
  • Number of vertices: 4                           

Pentagon

A pentagon is any five-sided polygon and the sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting regular pentagon is called a pentagram.
                                       
Properties:

  • Area: ½ × Perimeter × Apothem (A perpendicular line segment from the center of pentagon to aside.)
  • Perimeter: 5 x side
  • Number of vertices: 5
  • Number of edges: 5
  • Internal angle: 108°
  • Line of symmetry: 5

Hexagon

Hexagon is a polygon with six sides and opposite sides are parallel. The sum of the total internal angles of any simple hexagon is 720 degrees.
when all angles and length of sides are equal it is called a regular otherwise irregular hexagon.
                                     
 Properties:                       

  • Area: 3√3/2 × (side)²
  • Perimeter: 6 x side
  • Number of vertices: 6
  • Number of edges: 6
  • Internal angle: 120
  • Line of symmetry: 6

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