Geometry

Geometry is the branch of mathematics that deals with shapes, angles, dimensions and sizes of a variety of things we see in everyday life. Geometry is derived from Ancient Greek words – ‘Geo’ means ‘Earth’ and ‘metron’ means ‘measurement’. In Euclidean geometry, there are two-dimensional shapes and three-dimensional shapes.  

In a plane geometry, 2d shapes such as triangles, squares, rectangles, circles are also called flat shapes. In solid geometry, 3d shapes such as a cube, cuboid, cone, etc. are also called solids. The basic geometry is based on points, lines and planes explained in coordinate geometry. 

The different types of shapes in geometry help us to understand the shapes day to day life. With the help of geometric concepts, we can calculate the area, perimeter and volume of shapes. 

Table of Contents:

• Definition

• Branches

• Formulas

• Plane Geometry

  • Points

  • Lines

  • Angle

  • Polygon

  • Circle

• Solid Geometry

  • Edges

  • Faces

  • Vertices

Geometry Definition

Geometry is the study of different types of shapes, figures and sizes in Maths or in real life. In geometry, we learn about different angles, transformations and similarities in the figures. 

The basics of geometry depend on majorly point, line, angles and plane. All the geometrical shapes are based on these basic geometrical concepts.

Branches of Geometry

The branches of geometry are categorized as:

• Algebraic geometry

• Discrete geometry

• Differential geometry

• Euclidean geometry

• Convex geometry

• Topology

Algebraic Geometry

It is a branch of geometry studying zeros of the multivariate polynomial. It includes linear and polynomial algebraic equations that are used for solving the sets of zeros. The application of this type comprises Cryptography, string theory, etc.

Discrete Geometry

It is concerned with the relative position of simple geometric objects, such as points, lines, triangles, circles etc.

Differential Geometry

It uses techniques of algebra and calculus for problem-solving. The various problems include general relativity in physics etc.

Euclidean Geometry

The study of plane and solid figures based on 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, Crystallography etc.

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.

Topology

It is concerned with the 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, nets, proximal continuity, proximity spaces, separation axioms, and uniform spaces.
Also, check: Analytic geometry

Geometry Formulas

To calculate the area, perimeter, volume and different measures using length, breadth and height of different geometrical figures, geometry formulas are important. By using the formulas, we can calculate the measures easily. There are loads of formulas to learn in Geometry. Here, you are provided with all the important geometry formulas in the PDF. Download the below-given PDF to learn and solve all the problems related to geometry.

Geometry Formulas:-Download PDF Here

Plane Geometry (Two-dimensional 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 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.

The important terminologies in plane geometry are:

• Point

• Line

• Angles

Point

A point is a precise location or place on a plane. A dot usually represents them. It is important to understand that a point is not a thing, but a place. Also, note that a point has no dimension; preferably, 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 are 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 Geometry

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 (or Sharp angle) 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 than 180 degrees.

Right Angle – An angle of 90 degrees.

Straight Angle – An angle of 180 degrees is a straight angle, i.e. the angle formed by a straight line

Polygons in Geometry

A plane figure that is bounded by a finite chain of straight line segments 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 –

Types of Polygon

The types of polygons are: 

• Triangles

• Quadrilaterals

• Pentagon

• Hexagon

• Heptagon

• Octagon

• Nonagon

• Decagon

Polygon type

Definition & Property

Types

(i) Triangle –

A 3-sided polygon whose sum of internal angles 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 and vertices 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 parallel.

(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 ten angles.

–

In the below figure, we can see the different types of polygons.

Circle in Geometry

A Circle is a simple closed shape. From a certain point called the centre, all points of a circle are of same consistent distance, i.e. the curve traced out by a point that moves so that its distance from the centre is constant.

Similarity and Congruency in Geometry

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 (Three-dimensional 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).

Solid geometry is the study of three dimensions in Euclidean space. The objects which are around us are 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

• Vertices

Go through these terms in detail for different geometric shapes here.

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 meet in the straight line is called edge. Following are the list of edges for the different solid shapes:

Solid Shapes

No. of. Edges

Triangular Prism

9

Cube

12

Rectangular prism

12

Pentagonal Prism

15

Hexagonal Prism

18

Triangular Pyramid

6

Square Pyramid

8

Pentagonal Pyramid

10

Hexagonal Pyramid

12

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:

Solid Shapes

No. of. Faces

Triangular Prism

5

Cube

6

Rectangular prism

6

Pentagonal Prism

7

Hexagonal Prism

8

Triangular Pyramid

4

Square Pyramid

5

Pentagonal Pyramid

6

Hexagonal Pyramid

7

Vertices

A vertex is defined as the point where the edges of the solid figure meet at each other. In other words, 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:

Solid Shapes

No. of. Vertices

Triangular Prism

6

Cube

8

Rectangular prism

8

Pentagonal Prism

10

Hexagonal Prism

12

Triangular Pyramid

4

Square Pyramid

5

Pentagonal Pyramid

6

Hexagonal Pyramid

7

Geometry Related Articles

2D Shapes

3D Shapes

Acute Angle

Acute Angle Triangle

Annulus

Area Of Shapes

Cardioid

Centroid of a Triangle

Circles in Maths

Collinear Points

Cone

Congruence of Triangles

Conic Sections

Cube

Cylinder

Dodecagon

Ellipse

Equilateral Triangle

Faces, Edges and Vertices: 3D Shapes

Frustum Of A Cone

Frequency Polygons

Geometry Symbols

Heptagon

Hyperbola

Isosceles Triangle

Lines

Locus

Obtuse Angle

Orthocenter

Parallelepiped

Parallelogram

Perimeter

Polygon

Prism

Pyramid

Quadrilateral

Rectangle

Rhomboid

Rotational Symmetry

Scalene Triangle

Sector Of A Circle

Solids

Sphere

Square

Triangles

Trigonometry

Trigonometry Graphs

Types Of Angles

Video Lesson

Combination to form Geometry Models

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Frequently Asked Questions of Geometry

Q1

How geometry is used?

Geometry is used in configuring and measurement of different shapes and sizes, such as circles, squares, cubes, triangles, etc.

Q2

What are all the Geometry applications?

Geometry has wide usage in daily life. Its major applications are in the field of constructions, such as the construction of building, roads, dams, bridges, etc. and also in surveying, mapping, navigation. In software industries, it is used in graphics designing, gaming, animations, etc. In the medical field as well, for CT scanning and MRI’s geometry concepts are used.

Q3

Why geometry is important?

In our lives, geometry is important to understand the various shapes and measure them based on area and volumes.

Q4

What geometry means?

Geometry means the measurement of Earth, as it derived from the ancient Greek words, ‘geo’ means Earth and ‘metron’ means measurement. So any object which is present on earth has a shape which is defined by geometry.

Arithmetic

Arithmetic is one of the oldest and elementary branches of mathematics, originating from the Greek word ‘Arithmos’, which means ‘number’. Arithmetic involves the study of numbers, especially the properties of traditional operations, such as:

• Addition

• Subtraction

• Multiplication

• Division

The arithmetic operators based on these operations are ‘+’, ‘-’, ‘×’ and ‘÷’. Let us learn here all the important topics of arithmetic with examples.

What is Arithmetic?

Arithmetic is the fundamental of mathematics that includes the operations of numbers. These operations are addition, subtraction, multiplication and division. Arithmetic is one of the important branches of mathematics, that lays the foundation of the subject ‘Maths’, for students.

History of Arithmetic

The Fundamental principle of number theory was provided by Carl Friedrich Gauss in 1801, according to which, any integer which is greater than 1 can be described as the product of prime numbers in only one way. Arithmetic is another name given to number theory. The four elementary operations in arithmetic are addition, subtraction, multiplication and division. All these operations are discussed in brief here.

Arithmetic operations

The basic operations under arithmetic are addition and subtraction, division and multiplication, although the subject involves many other modified operations.

Addition (+)

Addition is among the basic operations in arithmetic. In simple forms, addition combines two or more values into a single term, for example: 2 + 5 = 7, 6 + 2 = 8, where ‘+’ is the addition operator.

The procedure of adding more than two values is called summation and involves methods to add n number of values.

The identity element of addition is 0, which means that adding 0 to any value gives the same result. The inverse element of addition is the opposite of any value, which means that adding the opposite of any digit to the digit itself gives the additive identity. For instance, the opposite of 5 is -5, therefore 5 + (-5) = 0.

Examples of addition:

• 8 + 10 = 18

• 12 + 5 = 17

Subtraction (−)

Subtraction can be labelled as the inverse of addition. It computes the difference between two values, i.e., the minuend minus the subtrahend. The operator of subtraction is (-).

If the minuend is greater than the subtrahend, the difference is positive. If the minuend is less than the subtrahend, the result is negative, and 0 if the numbers are equal.

For example,

• 4  – 3 = 1

• 3 – 4 = -1

Multiplication (×)

Multiplication also combines two values like addition and subtraction, into a single value or product. The two original values are known as the multiplicand and the multiplier, or simply both as factors.

The product of a and b is expressed as a·b or a × b, where ‘×’ is the multiplication operator. In software languages wherein only characters are used that are found in keyboards, it is often expressed as, a*b (* is called asterisk).

For example,

• 4 × 5 = 20

• 2 × 3 = 6

Division (÷)

The division is the inverse of multiplication. The operator used for the division method is ‘÷’ or ‘/’ sometimes. It computes the quotient of two numbers, the dividend that is divided by the divisor.

The quotient is more than 1 if the dividend is greater than the divisor for any well-defined positive number else, it is smaller than 1.

For example,

• 10 ÷ 2 = 5

• 9 ÷ 3 = 3

What is Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers, where the difference between one term and the next is a constant. For example, 1, 4, 7, 10, 13, 16, 19, 22, 25, … is an arithmetic sequence with common difference equal to 3. It is also termed arithmetic progression and is commonly represented as:

a, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1) d

Where, 

a = first term

d = common difference between the terms

n = number of terms

Arithmetic Solved Problems

Question 1: The sum of the two numbers is 50, and their difference is 30. Find the numbers.

Solution: Let the numbers be x and y. Now, as per the given situation,

x + y = 50……………………(i)

and x – y = 30………………(ii)

We can write, x = 50-y, from eq.(i),

Therefore, putting the value of x in eq(ii), we get,

50 – y – y = 30

50 -2y = 30

2y = 50-30= 20

y = 20/2 = 10

and x = 50 – y = 50-10 =40

Therefore, the two numbers are 40 and 10.

Question 2: Solve 25 + 5 (27 ÷ 3) – 9.

Solution: 25 + 5(27 ÷ 3) – 9

⇒  25 + 5(9) – 9

⇒ 25 + 45 – 9

⇒ 70 – 9 = 61

Practice Questions on Arithmetic

1. Solve: 4 + 3 x 10 – 1

2. Solve: 246 x 132

3. Solve: 9/10 – 3/20

4. 65 is what percent of 500?

Related Articles

Natural Numbers

Even And Odd Numbers

Laws of exponents

Multiplication and division

Decimals

Comparing Fractions

Percentages

Ratios and Proportion

Like and Unlike Fractions

Surds

Sum of n terms

Integers

Geometric Sequence

Fundamental Theorem

Fraction to decimal

Variable

Number Line

Whole Number

Even and odd numbers

Word problems on Decimals

Integers on a number line

Frequently Asked Questions

Q1

What is Arithmetic in Maths?

Arithmetic is one of the branches of mathematics which deals with different types of numbers like odd numbers, whole numbers, even numbers, etc. and their basic operations involve addition, subtraction, multiplication, and division.

Q2

What are the arithmetic operators?

The four basic arithmetic operators are addition (+), subtraction (-), multiplication (×) and division (÷).

Q3

What are the Properties of Operation in Arithmetic?

There are four main properties of operations which include:

• Commutative Property

• Associative Property

• Distributive Property

• Additive Identity

Q4

What is the Order of Operations?

The BODMAS or PEMDAS rule is followed to order any operation involving +, −, ×, and ÷. The order of operation is:

B: Brackets

O: Order

D: Division

M: Multiplication

A: Addition

S: Subtraction