Geometry is derived from two Latin words, Geo +Metron, Meaning is earth & measurement. Thus it is
concerned with the properties and relations of points, lines, surfaces, solids,
and higher dimensional analogy. It is the most practical branch of mathematics that deals with shapes and
sizes of figures and their properties. We study geometry to find the Length, Area, Volume of
different Plane and Solid figures which are present around us in this world and
to know better about them. Knowledge of coordinate Geometry provides many fundamental skills and helps us to improve
problem-solving skill, logical skill, analytical reasoning skill and so on.
Types of Geometry:
(i) Algebraic Geometry – It is a branch of
geometry studying zeros of the multivariate polynomial. It includes linear and
polynomial algebraic equation used for solving the sets of zeros. The
application of this type includes Cryptography, string theory, etc.
(ii) Discrete Geometry –It is concerned with the relative position of
simple geometric object, such as points, lines, triangles, circles etc.
(iii) Differential Geometry – It uses techniques
of algebra and calculus for problem-solving. The various problems include
general relativity in physics etc.
(iv) Euclidean Geometry – It is the study of
plane and solid figures on the basis of axioms and theorems including points,
lines, planes, angles, congruence, similarity, solid figures. It has a wide
range of applications in Computer Science, Modern Mathematics problem solving,
etc.
(v) Convex Geometry – It Includes convex
shapes in Euclidean space using techniques of real analysis. It has application
in optimization and functional analysis in number theory.
(vi) Topology –It is concerned with properties of space
under continuous mapping. Its application includes consideration of
compactness, completeness, continuity, filters, function spaces, grills,
clusters and bunches, hyperspace typologies, initial and final structures,
metric spaces, proximal continuity, proximity spaces, separation axioms, and
uniform spaces.
Plane Geometry:
Plane Geometry deals with flat shapes which can be drawn on a piece of
paper. These include lines, circles & triangles of two dimensions. Plane
geometry is also known as a two-dimensional geometry. All the two-dimensional
figures have only two measures such as length and breadth. It does not deal
with the depth of the shapes. Some examples of plane figures are square,
triangle, rectangle, circle, and so on.
Point :A precise location or place on a plane. Usually represented by a dot. A point is an exact
position or location on a plane surface. It is important to understand that a
point is not a thing, but a place. It is important to note that a point has no
dimension rather it has the only position.
Line: The line is straight (no curves), having no thickness and extends in both directions without end (infinitely). It is important to note that it is the combination of infinite points together to form a line. In geometry, we have a horizontal line and vertical line which has x-axis and y-axis respectively.
Line Segment – If a line has a starting and an endpoint then it is
called a Line Segment.
Ray – If a line has a starting point and has no endpoint is
called Ray. Eg. Sun Rays
Angles:
In plane geometry, an angle is
the figure formed by two rays, called the sides of the angle, sharing a common
endpoint, called the vertex of the angle.
Types of Angle:
Acute Angle – An Acute angle (or Sharp angle) is an angle smaller
than a right angle but greater than 00 i.e. it can range between 00
– 900 .
Obtuse Angle – An Obtuse angle is more than 900 but is
less than 180 degrees.
Right Angle – An angle of 900.
Straight Angle – An angle of 1800 is a straight angle. Such as angle formed by a
straight line
Polygon
A plane figure that is bounded
by a finite chain of straight line segment closing in a loop to form a closed
polygonal chain or circuit.
The name ‘poly’ refers to
multiple & gon is a polygon with n
sides; for example, a triangle is a 3-gon polygon.
Sum of internal Angles of a
polygon = (n−2)×1800
Where n is number of sides.
Each internal angle of regular
polygon = { (n−2)×1800}÷ n
Sum of external Angles of a
polygon = 3600
Each external angle of regular
polygon= {3600}÷ n
Types of Polygon
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Polygon type
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Definition & Property
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Types
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(i) Triangle –
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A 3-sided polygon whose sum
of internal angles always sums to 180 degrees.
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(ii)Quadrilateral
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A 4-sided polygon with four
edges and four vertices.
Sum of internal angles is
360 degrees
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(iii) Pentagon
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A plane figure with five
straight sides and five angles
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(iv) Hexagon
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A plane figure with six
straight sides and six angles
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(v) Heptagon
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A plane figure with seven
sides and seven angles
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(vi) Octagon
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A plane figure with eight
straight sides and eight angles.
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(vii) Nonagon
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A plane figure with nine
straight sides and nine angles.
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(viii) Decagon
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A plane figure with ten
straight sides and ten angles.
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Circle:A Circle is a simple closed
shape. From a certain point called the centre, all points of a circle are of
same consistent distance ie. the curve traced out by a point that moves so that
its distance from the centre is constant.
Understanding Similarity and Congruence:
Similarity – Two figures are said to be similar if they have the
same shape or have an equal angle but do not have the same size.
Congruence – Two figures are said to be Congruent if they
have the same shape and size. Thus they are totally equal.
Solid Geometry:
Solid Geometry:
Solid Geometry deals with 3-dimensional objects like cubes,
prisms, cylinders & spheres. It deals with three-dimensions of the figure
such as length, breadth and height. But some solid solids do not have faces
(e.g. sphere). It is the study of three dimensions in the Euclidean
space. The objects which are around us are obviously a three-dimensional. All
the three-dimensional shapes are obtained from the rotation operation of 2D
shapes. The important attributes of 3D shapes are faces, edges, and vertices. Let
us discuss these terms in detail for different geometric shapes.
Edges:
Edges:
An edge is defined as the line
segment on the boundary that joins one vertex to the other vertex. It means
that it joins one corner point to the other. It forms the skeleton of 3D shapes.
In other words, it can be defined as the faces, that meets in the straight line
is called edge. Following are the list of edges for the different solid shapes:
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Solid Shapes
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No. of. Edges
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Triangular Prism
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9
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Cube
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12
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Rectangular prism
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12
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Pentagonal Prism
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15
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Hexagonal Prism
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18
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Triangular Pyramid
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6
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Square Pyramid
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8
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Pentagonal Pyramid
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10
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Hexagonal Pyramid
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12
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Faces: We know that all the geometric
shapes are made up of flat surface called faces. It is a flat surface enclosed
by the edges. For any three-dimensional shapes, the face should be a
two-dimensional figure. The list of the
number of faces for different solid shapes are given below:
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Solid Shapes
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No. of. Faces
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Triangular Prism
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5
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Cube
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6
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Rectangular prism
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6
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Pentagonal Prism
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7
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Hexagonal Prism
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8
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Triangular Pyramid
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4
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Square Pyramid
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5
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Pentagonal Pyramid
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6
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Hexagonal Pyramid
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7
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Vertices: A vertex is defined as the
point where the edges of the solid figure meet at each other. Or else, it can
be said that, the point where the adjacent sides of the polygon meet. The
vertex is the corners where the edges meet. The number of vertices for
different solid shapes in geometry is as follows:
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Solid Shapes
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No. of. Vertices
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Triangular Prism
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6
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Cube
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8
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Rectangular prism
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8
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Pentagonal Prism
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10
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Hexagonal Prism
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12
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Triangular Pyramid
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4
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Square Pyramid
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5
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Pentagonal Pyramid
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6
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Hexagonal Pyramid
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7
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