Archimedes and geometry
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Archimedes made important contributions to geometry through his method of discovery and treatise "The Method". His approach involved balancing cross-sections of figures against known figures using laws of leverage. He would present his method of discovery before giving a rigorous proof. "The Method" contained Archimedes' processes for discovering many results on areas and volumes mechanically. Some of his key propositions included finding the volume of a cylinder inscribed in a parallelepiped and showing that a parabolic segment is 4/3 the size of an inscribed triangle.
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Modern Geometry Topics
This powerpoint includes:
Triangles and Quadrangles
Definition, Types, Properties, Secondary part, Congruency and Area
Definitions of Triangles and Quadrangles
Desarguesian Plane
Mathematician Desargues and His Background
Harmonic Sequence of Points/Lines
Illustrations and Animated Lines.
Mathematical analysis of identical circles touching one another on the whole ...
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All the articles discussed & analysed here are related to all five platonic solids. A certain no. of the identical circles are touching one another on a whole (entire) spherical surface having certain radius then all the important parameters such as flat radius & arc radius of each circle, total surface area & its percentage covered by all the circles on the sphere have been easily calculated by using simple geometry & table for the important parameters of all five platonic solids by the author Mr H.C. Rajpoot. These parameters are very useful for drawing the identical circles on a spherical surface & for designing & modeling all five platonic solids having identical flat circular faces.Hcr's formula for regular spherical polygon
This formula holds good for all the regular spherical polygons. It is a very important formula (mathematical relation) applicable on any regular spherical polygon having each of its sides as an arc of the great circle on a spherical surface. It is of crucial importance to find out any of the four important parameters (i.e. radius of sphere, no. of sides, length of side, interior angle of polygon) if other three are given (known) for any regular spherical polygon. It also concludes that any three of the four parameters are self-sufficient to exactly represent a unique regular spherical polygon. A regular spherical polygon having three known parameters can be created or drawn only on a unique spherical surface of a radius which is easily found out by HCR's formula.
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The document discusses several concepts in circle geometry and trigonometry including:
(1) Converse circle theorems such as if two points subtend the same angle from an interval, then they are concyclic.
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(3) Relationships between trigonometry and geometry such as the circumradius, inradius, and sine rule.
history of trigonometry
Trigonometry is the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant. Trigonometry originated in ancient civilizations for practical geometry applications and was further developed by Greek mathematicians like Hipparchus and Ptolemy. Indian and later Islamic mathematicians made important contributions, including the first tables of sines and tangents. Trigonometry was an important tool for astronomy and passed to Europe during the Middle Ages, with major works by Menelaus and Regiomontanus.
Properties of circle
Some properties of tangents, secants and chords, Angles formed by intersecting chords, tangent and chord and two secants, Chords and their arcs, Segments of chords secants and tangents, Lengths of arcs and areas of sectors
Triangles
This document defines and explains different types of triangles based on their sides and angles. It discusses equilateral, isosceles, scalene, right, obtuse, and acute triangles. It also covers calculating the perimeter, area, altitude, median, angle bisector, and inscribed/circumscribed triangles. Formulas are provided for calculating the altitude, median, angle bisector, and area using different known properties of triangles. Sample problems are included at the end to test understanding.
Circles PPT
The document discusses the properties and history of circles. It defines a circle as a set of points equidistant from a center point. Some key facts:
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Three dimensional space dfs
When ρ, θ, and Φ are all constant, the set of points described is a single point in 3D space. The spherical coordinates (ρ, θ, Φ) uniquely specify a location in 3D analogous to Cartesian coordinates (x, y, z).
Math slide ok-11860
This document discusses cones, frustums, and their properties. It defines a cone as having a circular base and curved side ending at an apex point. A frustum is formed by cutting off the tip of a cone with a cut perpendicular to the height, forming two parallel circular bases. The document provides formulas to calculate the volume and surface area of cones and frustums. It gives examples of applying the formulas to solve for volume and surface area.
Maths sa 2 synopsis
The document provides information about topics in grade 9 math, including:
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2) Properties of quadrilaterals such as parallelograms, and how to classify them.
3) Finding areas of parallelograms, triangles, and how these shapes are related.
4) Properties of circles such as angles subtended by chords and arcs.
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Analytical geometry
Okay, here are the steps to solve this problem:
1) The sphere has a radius of 8 inches
2) To find the volume of a sphere we use the formula: V = (4/3)πr^3
3) Plugging in the values:
V = (4/3)π(8)^3
V = (4/3)π(512)
V = 1024π
So the volume of the spherical fish tank with a radius of 8 inches is 1024π cubic inches.
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Problems in-plane-geometry-Sharygin
This document is the preface and table of contents for a book titled "Problems in Plane Geometry" that contains over 600 geometry problems divided into two sections. The preface provides background on the book, noting it contains both simple and more challenging problems, with detailed solutions. The table of contents outlines the chapters and topics covered, including fundamental theorems, loci of points, triangles, circles, polygons, and inequalities. The book is a collection of puzzles aimed at demonstrating elegant techniques in elementary geometry.
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The document discusses circle geometry and properties of triangles. It contains three theorems about circles: 1) A circle drawn through the vertices of a right triangle with diameter the hypotenuse passes through the third vertex. 2) If two points subtend the same angle from an interval, they lie on the same circle. 3) If opposite angles of a quadrilateral are supplementary, it is cyclic. It also discusses the four centers of a triangle: the incenter, circumcenter, centroid, and orthocenter and their properties. Finally, it proves the relationship between trigonometric ratios and the sides of a triangle inscribed in a circle.
X2 T08 01 circle geometry
The document discusses circle geometry and properties related to circles. It contains two main sections: (1) Converse circle theorems which describe properties of circles related to right triangles, angles subtended at points on a chord, and cyclic quadrilaterals. (2) The four centers of a triangle - the incentre, circumcentre, centroid, and orthocentre and their properties. It also discusses the interaction between geometry and trigonometry through a property relating side lengths and sine of angles of a triangle to the diameter of its circumcircle. An example problem demonstrates properties of angles and similar triangles related to a variable point on a circle.
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The document discusses circle geometry and properties of triangles. It contains three theorems about circles: 1) A circle drawn through the vertices of a right triangle with the hypotenuse as diameter will be tangent to the third side. 2) If two points subtend the same angle from an interval on the same side, the four points are concyclic. 3) If opposite angles of a quadrilateral are supplementary, it is cyclic. It also discusses the four centers of a triangle: the incentre, circumcentre, centroid, and orthocentre and their properties. Finally, it proves the relationship between trigonometric ratios and the sides of a triangle if the triangle is inscribed in a circle.
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Archimedes and geometry
- 1. Some of the Most Known Contributions of Archimedes in Geometry Prepared By: Katherine Bautista BSE Math
- 2. Archimedes Method of Discovery He often presented his method of discovery of the theorem and/or his analysis of the situation before presenting a rigorous synthetic proof.
- 3. “The Method” Archimedes treatise, collection of the methods of discovery of several of his results. Greek monastery library in 1989
- 4. Contains his methods of discovery by mechanics of many important results on areas and volumes.
- 5. Essential Features of The Method The balancing of cross sections of a given figure against corresponding cross sections of a known figure, using the law of the lever. The assumption that figures are composed of their indivisible cross sections.
- 6. SOME PROPOSITIONS If a cylinder is inscribed in a rectangular parallelepiped with a square base, and if a plane is drawn through the center of the circle at the base of the cylinder and through one side of the square forming the top of the parallelepiped, then the segment of the cylinder cut off by this plane has a volume equal to 1/6 of the entire parallelepiped.
- 7. A segment of a parabola is 4/3 of the triangle inscribe in it. The surface of any isosceles cone excluding the base is equal to a circle whose radius is a mean proportional between the side of the cone[a generator] and the radius of the circle which is the base of the cone.
- 8. The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.
- 9. Given a circle with center O, and the tangent to it at a point A, it is possible to draw from O a straight line OPF, meeting the circle in P and the tangent in F, such that, if c be the circumference of any given circle whatever.
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- 11. Conic Sections Before Apollonius Menaechmus (4th Century B.C.E.)
- 12. Thanks for not listening…^^