Geometry

Geometry (from the Ancient Greek: geo- "earth", -metron "measurement") is a branch of mathematics, that is primarily concerned with the shapes and sizes of the objects, their relative position, and the properties of space. There are many postulates and theorems applied by the Greek mathematician Euclid, who is often referred to as the “Father of Geometry”. Let us explore all the important topics in Geometry.

Geometry is the branch of mathematics that relates the principles covering distances, angles, patterns, areas, and volumes. All the visually and spatially related concepts are categorized under geometry. There are three types of geometry:

• Euclidean
• Hyperbolic
• Elliptical

We study Euclidean geometry to understand the fundamentals of geometry. Euclidean Geometry refers to the study of plane and solid figures on the basis of axioms (a statement or proposition) and theorems. The fundamental concepts of Euclidean geometry include Points and Lines, Euclid’s Axioms and Postulates, Geometrical Proof, and Euclid’s Fifth Postulate. There are 5 basic postulates of Euclidean Geometry that define geometrical figures.

• A straight line segment is drawn from any given point to any other.
• A straight line is extended indefinitely in both directions.
• A circle is drawn with any given point as its center and any length as its radius.
• All right angles are congruent.
• Any two straight lines are infinitely parallel that are equidistant from one another at two points.

Euclid's Axioms:

Axioms or postulates are based on assumptions and have no proof for them. A few of Euclid's axioms in geometry that are universally accepted are:

• The things that are equal to the same things are equal to one another. If A = C and B = C then A = C
• If equals are added to equals, the wholes are equal. If A = B and C = D, then A + C = B + D
• If equals are subtracted, the remainders are equal.
• The coinciding things are equal to one another.
• The whole is greater than its part. If A > B, then there exists C such that A = B + C.
• The things that are double the same are equal to one another.
• The things that are halves of the same things are equal to one another.

Spherical geometry and hyperbolic geometry are the two non-Euclidean geometries. Non-Euclidean geometry differs in its postulates on the nature of the parallel lines and the angles in the planar space, as validated by Euclidean geometry.

• Spherical geometry is the study of plane geometry on a sphere. Lines are defined as the shortest distance between the two points that lie along with them. This line on a sphere is an arc and is called the great circle. The sum of the angles in the triangle is greater than 180º.
• Hyperbolic geometry refers to a curved surface. This geometry finds its application in topology. Depending on the inner curvature of the curved surface, the planar triangle has the sum of the angles lesser than 180º.

Euclidean geometry involves the study of geometry in a plane. A two-dimensional surface extending infinitely in both directions forms the plane. Planes are used in every area of geometry and graph theory. The basic components of planes in geometry are analogous to points, lines, and angles. A point is the no-dimensional basic unit of geometry. Points lying on the same line are the collinear points. A line is a uni-dimensional unit that refers to a set of points that extends in two opposite directions and the line is said to be the intersection of two planes. A line has no endpoints. It is easy to differentiate a line, line segment, and ray. Lines may be parallel or perpendicular. Lines may or not intersect.

Angles in Geometry

When two straight lines or rays intersect at a point, they form an angle. Angles are usually measured in degrees. The angles can be an acute, obtuse, right angle, straight angle, or obtuse angle. The pairs of angles can be supplementary or complementary. The construction of angles and lines is an intricate component of geometry. The study of angles in a unit circle and that of a triangle forms the stepping stone of trigonometry. Transversals and related angles establish the interesting properties of parallel lines and their theorems.

Plane Shapes in Geometry

The properties of plane shapes help us identify and classify them. The plane geometric shapes are two-dimensional shapes or flat shapes. Polygons are closed curves that are made up of more than two lines. A triangle is a closed figure with three sides and three vertices. There are many theorems based on the triangles that help us understand the properties of triangles. In geometry, the most significant theorems based on triangles include Heron's formula, The exterior angle theorem, the angle sum property, the basic proportionality theorem, the similarity and Congruence in Triangles, the Pythagoras Theorem, and so on. These help us recognize the angle-side relationships in triangles. Quadrilaterals are polygons with four sides and four vertices. A circle is a closed figure and has no edges or corners. It is defined as the set of all points in a plane that are equidistant from a given point called the center of the circle. Various concepts centered around symmetry, transformations in shapes, construction of shapes are the formative chapters in geometry.

Solid shapes in geometry are three-dimensional in nature. The three dimensions that are taken into consideration are length, width, and height. There are different types of solid figures like a cylinder, cube, sphere, cone, cuboids, prism, pyramids, and so on and these figures acquire some space. They are characterized by vertices, faces, and edges. The five platonic solids and the polyhedrons have interesting properties in Euclidean space. The nets of the plane shapes can be folded into solids.

Measurement in geometry ascertains the calculation of length or distance, the area occupied by a flat shape, and the volume occupied by the solid objects. Mensuration in geometry is applied to the computation of perimeter, area, capacity, surface areas, and volumes of geometric figures. Perimeter is the distance around the plane shapes, the area is the region occupied by the shape, volume is the amount of region occupied by a solid, and the surface area of a solid is the sum of the areas of its faces.

Analytical geometry is popularly known as coordinate geometry is a branch of geometry where the position of any given point on a plane is defined with the help of an ordered pair of numbers, or coordinates using the rectangular Cartesian coordinate system. The coordinate axes divide the plane into four quadrants. Identifying and plotting points will be a building block of visualizing the geometric objects on the coordinate plane. In the example below, point A is defined as (4,3) and Point B is defined as (-3,1).

The various properties of the geometric figures like straight lines, curves, parabolas, ellipse, hyperbola, circles, and so on can be studied using coordinate geometry. In analytical geometry, the curves are represented as algebraic equations, and this gives a deeper understanding of algebraic equations through visual representations. The distance formula, the section formula, midpoint formula, the centroid of a triangle, the area of the triangle formed by three given points, and the area of the quadrilateral formed by four points are determined using the known coordinates in the cartesian coordinate system. The equation of a straight line passing through a point, or two points, the angle between two straight lines are computed easily using the analytical geometry as they are generalized using formulas.

The three-dimensional geometry discusses the geometry of shapes in 3D space in the cartesian planes. Every point in the space is denoted by 3 coordinates, represented as an ordered triple (x, y,z) of real numbers.

Direction Cosines of a Line

If a straight line makes angles α, β and γ with the x-axis, y-axis, and z-axis respectively then cosα, cosβ, cosγ are called the direction cosines of a line. These are denoted as l = cosα, m = cosβ, and n = cosγ. For l, m, and n, l² + m² + n² = 1, direction cosines of a line joining the points P(\(x_1, y_1, z_1\)) and Q(\(x_2, y_2, z_2\)) are given as :\(\dfrac{x_2-x_1}{PQ}, \dfrac{y_2-y_1}{PQ}, \dfrac{z_2-z_1}{PQ}\),

where PQ = \(\sqrt{((x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\)

Direction Ratios of a Line

The directional ratios of a line are the numbers that are proportional to the direct cosines of the line. If l, m, n are the direction cosines, and a,b c are the direction ratios, then

l = \(\dfrac{a}{\sqrt{a^2+b^2+c^2}}\),

m = \(\dfrac{b}{\sqrt{a^2+b^2+c^2}}\) and

n = \(\dfrac{c}{\sqrt{a^2+b^2+c^2}}\).

Direction ratios of line joining the points P(\(x_1, y_1, z_1\)) and Q(\(x_2, y_2, z_2\)) are:

\((x_2-x_1),(y_2-y_1), (z_2 -z_1)\) or \((x_1-x_2),(y_1-y_2), (z_1 -z_2)\)

Skew lines in Geometry

The skew lines are the lines in space that are neither parallel nor intersecting, and they lie in different planes. The angle between two lines is cos θ = |\(l_1l_2 + m_1m_2 + n_1n_2\)| where θ is the acute angle between the lines.

Also Cos θ = |\(\dfrac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_1^2+b_1^2+c_1^2}}\)|

Equation of Line in 3-D Geometry

• Vector equation of the line passing through a point with the position vector \(\vec a\) and parallel to vector \(\vec b\) is \(\vec r = \vec a+ \lambda \vec b\)
• Cartesian equation of the line passing through the point (\(x_1, y_1, z_1\)) and direction cosines l, m, n is \(\dfrac{x -x_1}{l} =\dfrac{y -y_1}{m}= \dfrac{z -z_1}{n}\)
• Vector equation of the line passing through two points with the position vectors \(\vec a\) and \(\vec b\) is \(\vec r = \vec a+ \lambda(\vec b -\vec a)\)
• Cartesian equation of the line passing through the points (\(x_1, y_1, z_1\)) and (\(x_2, y_2, z_2\)) is \(\dfrac{x -x_1}{x_2 -x_1} =\dfrac{y -y_1}{y_2 -y_1}= \dfrac{z -z_1}{z_2 -z_1}\)

Angle Between Two Lines

Angle between intersecting lines drawn parallel to each of the skew lines is the angle between skew lines. If θ is the angle between \(\vec r = \vec a_1+ \lambda \vec b_1\) and \(\vec r = \vec a_2+ \lambda \vec b_2\), then cos θ = |\(\dfrac{\vec b_1 . \vec b_2}{|\vec b_1| |\vec b_2|}\)|

Also Check:

• Points and Lines
• Essence of Geometric Constructions
• Slope of a line
• Euclidean Distance Formula
• All Circle Formulas

What is Geometry?

Geometry is the branch of mathematics that studies the shape, size, patterns, angle positions, dimensions, and properties of the objects around us and the spatial relationships among the objects.

What are The Three Types of Geometry?

The three types of geometry are Euclidean, Hyperbolic, and Elliptical Geometry.

Who is The Father of Geometry?

The father of Geometry is Euclid, a Greek mathematician. His book 'Elements' is the most influential work in maths serves as the fundamentals of geometry. It is a collection of propositions and postulates. The book consists of 13 volumes and it was second to the Bible in publications.

What are The Five Basic Postulates of Euclidean Geometry?

The five basic postulates are meant to be the fundamentals of geometry learned in the formative years.

• A straight line segment may be drawn from any given point to any other.
• A straight line may be extended indefinitely in both directions.
• A circle may be drawn with any given point as its center and any length as its radius.
• All right angles are congruent.
• Any two straight lines are infinitely parallel that are equidistant from one another at two points.

What is LW in Geometry?

LW is the formula used in geometry to find the area of a rectangle. Length and width are the parameters of a rectangle. The product of length and width is the area of the rectangle. Area of a rectangle = LW square units.

How Do You Use Euclidean Geometry?

Euclidean geometry studies the basic and complex geometric structures that are both plane shapes and solid shapes. Thus Euclidean geometry is used in art and architecture, computer science, astronomy, and other fields of mathematics. Euclidean geometry considers the study of points, lines, angles, and similarity and congruence in shapes, their patterns, and their transformations. Thus it has its practical applications in our day-to-day life.

Why is it Called Hyperbolic Geometry?

The 5th Euclidean postulate on parallel lines is not validated by the hyperbolic geometry. Hyperbolic geometry illustrates three key points that differ from Euclidean geometry.

• Two parallel lines converge in one direction and diverge in the other.
• The sum of angles in a triangle is less than 180º.
• Similar polygons of different areas don't exist.

What Is The Difference Between Euclidean and Non-Euclidean Geometry?

Euclidean Geometry is the study of the geometry of flat shapes on a plane, while non-Euclidean geometry aims at studying curved surfaces.

What are the 2 Types of Geometry?

Two types of geometry are plane geometry and solid geometry. Plane geometry deals with two-dimensional shapes and planes (x-axis and y-axis), while solid geometry deals with three-dimensional objects and 3D planes. These are the two types of geometry.

What Is The Difference Between Solid and Plane Geometry?

The study of shapes and objects that have length, width, and height is called the solid geometry, whereas the study of the shapes that have no thickness and can be represented only on a two-dimensional plane surface is called plane geometry.

What is Elliptic Geometry Used For?

Elliptic geometry studies the geometric structures that have curved surfaces. We use elliptic geometry to find the distance between the heavenly bodies in space, to calculate the distance between the places on the earth.

How do you Teach Geometry?

Teaching geometry is not an easy task that can be done with a textbook and drawing some shapes on the board. It requires a range of activities that demand learners' involvement to understand this concept with much more clarity. Some of the teaching-learning activities are listed below:

• Visualization- It involves learning through real-life experiences. We can take learners outside the classroom and help them to observe different shapes of objects, x-axis, and y-axis on the floor or any other flat surface, etc.
• Demonstration- It includes bringing some real-life objects to represent a concept in geometry. For example, it is always better to bring dice or any other object that represents a cubical shape to make learners understand the properties of a cube.
• DIY activities- We can ask learners to participate in some Do-It-Yourself activities that help them to work and play around with shapes and other concepts in geometry.
• Introducing Scientific Concepts- After all these activities, we can introduce the names and properties used in geometry using scientific terms. It includes introducing the terms like a cartesian plane, polyhedrons, quadrilaterals, etc.

What is Geometry Divided Into?

Geometry is the branch in mathematics that is further divided into various sub-branches that are given in the list below:

• Euclid’s Geometry
• Lines
• Angles
• Plane Shapes
• Solid Shapes
• Coordinate Geometry
• Vectors

What are the Basics of Geometry?

The basics of geometry are the proper understanding of points, lines, and planes. It then helps in building all other concepts in geometry that are based on these basic concepts.