euclidean geometry Euclidean geometry is the study of 2-Dimensional geometrical shapes and figures. Euclidean geometry is based on different axioms and theorems. The word geometry is derived from the Greek words ‘geo’ meaning Earth and ‘metrein’ meaning ‘To measure’. Thus, geometry is the measure of the Earth or various shapes present on the Earth. Euclidean geometry as the name suggests was first used by the famous Greek mathematician Euclid. He described the geometry of flat objects in his book “Elements” and was the pioneer in this field. He gives various axioms or postulates that are obvious universal truths, (but they can not be proved by usual means). Euclidean Geometry is considered an axiomatic system, where all the theorems are derived from a small number of simple axioms. Since the term “Geometry” deals with things like points, lines, angles, squares, triangles, and other shapes, Euclidean Geometry is also known as “plane geometry”. It deals with the properties and relationships between all things. Plane Geometry 1. Congruence of triangles 2. Similarity of triangles 3. Areas 4. Pythagorean theorem 5. Circles 6. Regular polygons 7. Conic sections Solid Geometry 1. Volume 2. Regular solids A solid has shape, size, and position and can be moved from one place to another. Its boundaries are called surfaces. They separate one part of space from another. They are assumed to have no thickness. The boundaries of surfaces are made up of curves or straight lines. These lines or curves are in turn made up of points. From solid to the line, at each step, we lose one extension which is defined as dimension. Euclid’s Definitions 1. A point is that which has no part. 2. A line is breadthless. 3. The ends of a line are points. 4. A straight line is a line that lies evenly with the points on itself. 5. A surface is that which has length and breadth only. 6. The edges of a surface are lines. 7. A plane surface is a surface that lies evenly with straight lines on it. The table below mentions the theorems that were proved by Euclid. Plane Geometry Congruence of Triangles Similarity of Triangles history of euclidean geometry Geometry is widely used in the History of mankind for various purposes, artifacts and sites are found that state that ancient humans have a very deep knowledge of geometry. In one of the earliest cities of the world such as in Harappa and Mohenjo-Daro historians have found that these cities are planned with proper drainage systems and other facilities. This proves that they have a deep understanding of Geometry. Also, one of the wonders of the world the Pyramid of Giza in Egypt, can not be built without the knowledge of Geometry. euclid’s geometry These ancient civilizations used geometry mostly for practical purposes and there was little emphasis on the reasoning behind the statements. Euclid thought of geometry as an abstract model of the world. The idea of the points, lines, and shapes was derived from what was seen around in the real world. He studied real-world objects to formalize the concept of solid. Theorem Proved Two triangles are congruent if they are similar in shape and size. Two triangles are similar in shape but differ in size. Areas Area of a plane shape can be measured by comparing it with a unit square. Pythagorean Theorem Pythagorean theorem helps in calculating the distance in different situations for Geometric shapes. Circles Equal chord determines equal angles and vice versa in a circle. Regular Polygons Conic Section Solid Geometry Volume Regular Solids Regular Polygons are equal in sides and angles. Conic sections include Ellipse, Parabola, and Hyperbola. Theorem Proved Volume of a shape can be calculated. The existence of Platonic Solids. examples of euclidean geometry There are various examples of Euclidean geometry and some of the common examples of Euclidean Geometry include the geometry of, โ Angles โ Circles Euclid defines angles as the inclination of one straight line on another. Similarly, a circle is defined as a plane figure in which all the points on the boundary are at a constant distance from the fixed point (center). Euclidean Geometry Non-Euclidean Geometry 1. Deals with flat figures in 2-dimension 1. Deals with spherical geometry and 2. Only one line parallel to another line hyperbolic geometry in 3-D and passing through a fixed point 2. Multiple lines in 3-dimensions which are at a constant distance from a line and passes through a line. properties of euclidean geometry 1. Euclidean Geometry is the study of plane 2-dimensional figures. 2. Euclidean Geometry defines a point, a line, and a plane. 3. It says that a solid has shape, size, and position, and it can be moved from one place to another. 4. The sum of the interior angles of the triangle is 180 degrees. 5. Parallel lines never intersect each other. 6. The shortest distance between two lines is always the perpendicular distance between them. euclid’s axiom Axioms – universally true and needs no proof 1. Things which are equal to the same thing are equal to one another. 2. If equals are added to equals, the wholes are equal. 3. If equals are subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part. 6. Things which are double of the same things are equal to one another. 7. Things which are halves of the same things are equal to one another. euclid’s postulates 1. A straight line may be drawn from any one point to any other point. Axiom: Given two distinct points, there is a unique line that passes through them. 2. A terminated line can be produced indefinitely. Euclid terms of a line segment as a terminated line. 3. A circle can be drawn with any center and any radius. Any circle can be drawn from the end or start point of a circle and the diameter of the circle will be the length of the line segment. 4. All right angles are equal to one another. All the right angles (i.e. angles whose measure is 90°) are always congruent to each other i.e. they are equal irrespective of the length of the sides or their orientations. 5. If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles. 90° 90° 360° = 1 4 ๐ด = 1 4 × π๐ radii and chords 45° 45° 360° = 1 8 ๐ด = 1 8 × π๐ 30° 30° 360° = 1 12 ๐ด = 1 12 × π๐ ๐ด = θ 360 × π๐ โ A circle is the set of all points in the plane that are a fixed distance (the radius) from a fixed point (the center). โ Any interval joining a point on the circle to the center is called a radius. By the definition of a circle, any two radii have the same length. โ An interval joining two points on the circle is called a chord. โ A chord that passes through the center is called a diameter. Since a diameter consists of two radii joined at their endpoints, every diameter has a length equal to twice the radius. โ An arc whose measure is less than 180° is called a minor arc. An arc whose measure is greater than 180° is called a major arc. An arc whose measure equals 180° is called a semicircle since it divides the circle in two. โ Subtended angles are any of the several angles that may be created by merging the arcs' ends in a circle. โ A chord AB of a circle divides the circle into two segments. The two segments are called a major segment (the larger one) and a minor segment (the smaller one). โ A sector is a part of a circle made of the arc of the circle along with its two radii. It is a portion of the circle formed by a portion of the circumference (arc) and radii of the circle at both endpoints of the arc. θ = angle so 1 2π๐ = ๐ด๐๐๐ = = 180° θ 360° × 2π๐ × π๐ Fraction of circle area 180° 360° = 1 2 Area rule ๐ด = 2 1 2 2 2πθ × ๐ ( 360° − ๐ ๐๐ θ) the earth as a sphere θ 360° θ 360° θ 180° 2 Where a chord subtends an angle θ at the center of a circle of radius ๐, the area of the minor segment is given by: ๐ = radius ∴ ๐ = Angle = 2 areas of segments In any circle, the length of an arc (๐) is proportional to the angle (θ) it creates (subtends) at the center. For a circle of radius ๐, the circumference will be 2π units. ๐กโ๐ ๐๐๐๐๐กโ ๐๐ ๐กโ๐ ๐๐๐ ๐๐๐๐๐ข๐๐๐๐๐๐๐๐ θ 360° θ measuring arc length ๐ = length of an arc 2 spherical geometry 1 2 × π๐ 2 โ A great circle is the largest circle that can be drawn on the surface of a sphere, as the center of the great circle is also the center of the sphere. The radius of a great circle on Earth will therefore be 6 400 kilometers which is the same as the radius of the Earth sphere. As shown above, the equator and a circle drawn through both the North and South poles are great circles. The meridians of longitude are all great circles. โ A small circle does not have its center at the center of the sphere. The radius of a small circle will be LESS than the radius of the sphere. For Earth, it means that the radius of a small circle will be less than 6 400 kilometers. The parallels of latitude (apart from the Equator) are all small circles โ Latitude is a measure of the position of a point on the earth’s surface in terms of degrees north or south of a baseline – the equator. These values vary from 90°S to 90°N. The latitude of the Equator is 0°. โ Every plane perpendicular (at right angles) to the earth’s axis cuts the surface of the earth in a circle called a parallel of latitude. โ The parallel of latitude formed by a plane passing through the centre of the earth and equidistant from both the North and South Poles is called the equator. All other parallels of latitude are small circles, each having a radius smaller than that of the earth, i.e. less than 6400 km. As the parallels run east and west, places having the same latitude will be due east or due west of each other. โ Latitude is determined by measuring the angle of elevation of the sun at noon, and adding or subtracting an allowance for the time of year. Using the angle only is correct for just two days of the year - the spring and autumnal equinoxes. In midwinter, subtract 23.45° and at midsummer, add 23.45°. These allowances are due to the fact that the Earth is tilted at this angle as it orbits the sun. Books of correction factors for every day of the year (almanacs) were used for many years until global positioning satellites were used. โ In the same way that latitude fixes a place north or south of the equator, longitude fixes a place east or west of the prime meridian - also known as the Greenwich meridian which passes through Greenwich near London, UK โ Each great circle passing through the north and south poles has the earth’s axis as its diameter. Each great circle is divided by the poles into two semi-circles called meridians of longitude. As the meridians run north and south, places having the same longitude will be due or due south of each other. Distance east or west of Greenwich is measured as an angle between the plane containing the Greenwich meridian and the plane containing the meridian passing through the point whose longitude is required.
