# Geometry

Geometry is derived from two Latin words, geo + metron meaning earth & measurement. Thus it is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.

Why we Study about Geometry?

To find Length, Area, Volume of different Plane and Solid figures of use which are present around us in this world and to know better about them.

#### Contemporary geometry has many subfields which are as follows-

(i) Algebraic Geometry – is a branch of geometry studying zeros of multivariate polynomial. It include linear and polynomial algebraic equation used for solving about the sets of zeros.The application of this type include Cryptography, string theory etc.

(ii) Discrete Geometry – is concerned with relative position of simple geometric object, such as points, lines , triangles, circles etc.

(iii) Differential Geometry – uses techniques of algebra and calculus for problem solving. The various problem include general relativity in physics etc.

(iv) Euclidean Geometry – 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 application in Computer Science, Modern Mathematics problem solving, Crystallography etc.

(v) Convex Geometry – includes convex shapes in Euclidean space using techniques of real analysis. It has application in optimization and functional analysis in number theory.

(vi) Topology – 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 topologies, initial and final structures, metric spaces, metrization, nets, proximal continuity, proximity spaces, separation axioms, and uniform spaces.

Another way of classification of Geometry include Plane and Solid Geometry.

Plane Geometry – This wing of Geometry deals with flat shapes which can be drawn on piece of paper. These includes lines, circles & triangles of two dimensions.

Soid Geometry – It deals with 3-dimensional objects like cubes, prisms, cylinders & spheres.

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 only position.

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.

• Line Segment – If a line has a starting and an end point then it is called as Line Segment.
• Ray – If a line has a starting point and has no end point is called as Ray.

Eg. Sun Rays

### Angles – In planar 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 meaning Sharp is an angle smaller than a right angle ie. it can range between 0 – 90 degrees.

Obtuse Angle – An Obtuse angle is more than 90 degrees but is less that 180 degrees.

Right Angle – An angle of 90 degree.

Straight Angle – An angle of 180 degree 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. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon polygon.

General Formula for Sum of internal Angles of a polygon –

Sum of internal Angles of a polygon = $(n-2)\times 180$<

### Different types of Polygon –

 Polygon type Definition & Property Types (i) Triangle – A 3-sided polygon whose sum of internal angle always sums to 180 degrees. Equilateral Triangle – Has 3 equal sides and angles. Isosceles triangle –  Has 2 equal sides and angles. Scalene triangle – Has all the 3 unequal sides and angles. (ii) Quadrilateral A 4-sided polygon with four edges and four vertices. Sum of internal angles is 360 degrees Square – Has 4 equal sides which are at right angles. Rectangle – Has equal opposite sides and all angles are at right angles. Parallelogram – Has two pairs of parallel sides. The opposite sides & opposite angles are equal in measure. Rhombus – Has all the four sides to be of equal length. However they do not have its internal angle to be 90 degrees Trapezium – Has one pair of opposite sides to be equal. (iii) Pentagon A plane figure with five straight sides and five angles (iv) Hexagon A plane figure with six straight sides and six angles (v) Heptagon A plane figure with seven sides and seven angles (vi) Octagon A plane figure with eight straight sides and eight angles. (vii) Nonagon A plane figure with nine straight sides and nine angles. (viii) Decagon A plane figure with ten straight sides and angles.

## Circle –

A Circle is a simple closed shape. From a certain point called 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 centre is constant .

Understanding Similarity and Congruence –

Similarity – Two figures are said to be Similar if they have same shape or have equal angle but do not have same size.

Congruence –  Two figures are said to be Congruent if they have same shape and size. Thus they are totally equal.

 Related Articles Lines Angles Polygon Triangles Quadrilateral Circles 3 Dimensional shapes Rectangles Parallelogram

#### Practise This Question

An exact location in space is called a _______