The Ultimate Guide to Geometry

Sunil Jakhar

Summary

Welcome to this ultimate guide to geometry. We have complied a refined selection of the most important information you need to know. This guide will help to grasp the foundations of geometry, and the history that led us to discover so much about the world in which we live.

KEY POINTS

✔ Geometry is an ancient study of numbers
✔ Its amazing
✔ Simple calculations through shape and form

## Introduction

The Guide to Geometry provides you with everything you need to know to explore this fascinating subject. Geometry is the study of numbers in space. Alongside Arithmetic, pure number, it is one of the oldest branches of mathematics that studies the sizes, shapes, positions angles and dimensions of objects. In school, we learn geometry to measure the length, area, and volume of different 2D shapes (length and width) such as circles, triangles, squares or in 3D space (length, width & height), auch as spheres & cubes. Rooted in the Quadrivium of the 7 liberal Arts of Ancient Greece, this Guide takes on Euclidean geometry.

## History of Geometry

Geometry originates from the Ancient Greeks which translates into ‘Geo’ meaning ‘earth’ and ‘metron’, measurement. While Euclid of Alexandria is known as the father of geometry, he was not the first person who used geometry. The earliest recorded geometric images go as far back as ancient Babylonia from around 3000 BC. The famous Pythagoras Theorem of right angled triangles can be traced back to Ancient Egypt and Babylon 1500 years before Pythagoras and is even found in ancient scripture of India.

## Fundamentals

Geometry can be defined as a quantitative and qualitative study of space and forms. The most fundamental attributes about space/form include shape and size. Geometry is a deductive system. This means there is a logical development and a few simple, fundamental principles.

### Basic Forms

• A point is that which has no part.
• A line is breadth-less length.
• The ends of a line are points.
• A straight line is a line which lies evenly with the points on itself.
• A surface is that which has length and breadth only.
• The edges of a surface are lines.
• A plane surface is a surface which lies evenly with the straight lines on itself.
• A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
• And when the lines containing the angle are straight, the angle is called rectilinear.
• When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
• An obtuse angle is an angle greater than a right angle.
• An acute angle is an angle less than a right angle.
• A boundary is that which is an extremity of anything.
• A figure is that which is contained by any boundary or boundaries.
• A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another.
• And the point is called the center of the circle.
• A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.
• A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.
• Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.
• Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that has two of its sides alone equal, and a scalene triangle that has its three sides unequal.
• Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that has an obtuse angle, and an acute-angled triangle that has its three angles acute.
• Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
• Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

### Axioms

Axioms can be understood as self-evident assumption, as starting point for further explanation. The word comes from Greek axíōma and translates into ‘that which is evident.’

The fundamental principles of geometry are called Axioms. These are propositions accepted as true without mathematical proof.

Axioms are the foundation to understand the nature of space, its objects and properties through logical reasoning.

• To draw a straight line from any point to any point.
• To produce a finite straight line continuously in a straight line.
• To describe a circle with any center and radius.
• That all right angles equal one another.
• That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Apart from the axioms, geometry proceeds with certain definitions and a set of actions taken for grant. Together all of these are called the first principles of geometry

### Common Notions

Things which equal the same thing also equal one another.
If equals are added to equals, then the wholes are equal.
If equals are subtracted from equals, then the remainders are equal.
Things which coincide with one another equal one another.
The whole is greater than the part.

## Plane geometry

### Triangle

A triangle is a geometrical figure that is contained by three straight lines. Every triangle has three sides and three angles. It is most basic as well as the most important shape. The triangle serves as a basis to understand all other geometrical figures and their properties. The reason is that the triangle is the only rigid figure among all polygons as there is no scope for “wiggling”. Unlike all other higher polygons, a triangle can be completely defined using the least number of parameters.

Triangles are generally classified under four categories:

A triangle is a geometrical figure that is contained by three straight lines. Every triangle has three sides and three angles. It is most basic as well as the most important shape. The triangle serves as a basis to understand all other geometrical figures and their properties. The reason is that the triangle is the only rigid figure among all polygons as there is no scope for “wiggling”. Unlike all other higher polygons, a triangle can be completely defined using the least number of parameters.

Triangles are generally classified under four categories:

1. Equilateral: The triangles which have all sides of equal in length.
2. Isosceles: The triangles which have two of the sides of equal in length.
3. Scalene: The triangles which have all sides of different lengths.
4. Right Angled: The triangles which have one of the angles as the right angle. The side opposite to the right angle is referred to as the hypotenuse.

#### Congruence of Triangles

Geometrical objects are said to be congruent when they have the same shape and size. ‘Congruent’ can be understood as the geometrical equivalent of the common term ‘equal’. The necessary and sufficient conditions to decide congruence in case of triangles are as follows:

• SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.
• SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.
• ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.
• AAS (Angle-Angle-Side): If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent.
• RHS (Right-angle-Hypotenuse-Side): If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent.

#### Similarity of Triangles

Geometrical objects are said to be similar when they have the same shape but not necessarily the same size. All geometric figures that can be seen as merely scaled up or scaled down versions of each other can be called similar. All circles are similar irrespective of their size because they all have the same shape.

In case of triangles, the conditions to determine similarity are as follows:

1. If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar.
2. If corresponding sides of triangles are in proportion, then the triangles are similar.

## Polygon

A quadrilateral is a rectilinear figure bound by four straight lines called edges or sides. Quadrilateral also has four angles defined by the points where the edges meet (called vertex).

Trapezium: When none of the sides containing the quadrilateral are parallel.

Trapezoid: When two sides of the quadrilateral are mutually parallel it is called a trapezoid. If the non parallel sides are equal in that case the trapezoid is called isosceles.

Parallelogram: If the pair of opposite sides are parallel, such quadrilateral is called a parallelogram. Kinds of parallelogram:

Rectangle: When all the angles contained in the parallelogram are right angles it is called a rectangle.

Rhombus: If all the sides containing the parallelogram are equal to each other it is called a rhombus.

Square: A square is a unique parallelogram which has all sides equal as well as all angles as right angles. It can be considered a special case of rectangle as well as rhombus.

### Circle

A circle is a closed curve lying in a plane such that all of its points are at equal distance from a fixed point in the plane (called the center of the circle).

Radius: A straight line from the center to the circle.

Diameter: A straight line through the center, terminating at each end of the circle.

Arc: A portion of the circle is called an arc. An arc equal to half of the circle is called a semicircle. An arc less than the semicircle is called a minor arc while an arc greater than the semicircle is called a major arc.

#### Chords, Secants & Tangents

Secant: A line intersecting a circle in two places is referred to as a secant.

Chord: The portion of secant that is contained within the circle is called a chord.

Tangent: If a line intersects (touches) a circle at only one single point, it is called a tangent.

#### Inscribed polygons

An inscribed polygon is a polygon in which all vertices lie on a circle. The polygon is inscribed in the circle and the circle is circumscribed about the polygon. It is a polygon in a circle.

#### Circumscribed polygons

A circumscribed polygon is a polygon in which each side is a tangent to a circle.The circle is inscribed in the polygon and the polygon is circumscribed about the circle. It is a polygon outside the circle or in other words the circle is inside a polygon.

### Mensuration

Mensuration is the part of geometry that is concerned with ascertaining lengths, areas, and volumes. Mensuration concerns with the measurement of the geometric figures and their parameters like length, volume, shape, surface area, lateral surface area, etc.

#### Perimeter of polygons

Perimeter can be defined as the total length of the boundary of a geometrical figure.

#### Area of polygons

Area is the measure of the surface contained inside a geometrical figure. It is generally measured against the unit square, or say, how much surface in comparison with a square of unit side length.

## Solid Geometry

• A solid is that which has length, breadth, and depth.
• A face of a solid is a surface.
• A straight line is at right angles to a plane when it makes right angles with all the straight lines which meet it and are in the plane.
• A plane is at right angles to a plane when the straight lines drawn in one of the planes at right angles to the intersection of the planes are at right angles to the remaining plane.
• The inclination of a straight line to a plane is, assuming a perpendicular drawn from the end of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the end of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up.
• The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the intersection at the same point, one in each of the planes.
• A plane is said to be similarly inclined to a plane as another is to another when the said angles of the inclinations equal one another.
• Parallel planes are those which do not meet.
• Similar solid figures are those contained by similar planes equal in multitude.
• Equal and similar solid figures are those contained by similar planes equal in multitude and magnitude.
• A solid angle is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines, that is, a solid angle is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point.
• A pyramid is a solid figure contained by planes which is constructed from one plane to one point.
• A prism is a solid figure contained by planes two of which, namely those which are opposite, are equal, similar, and parallel, while the rest are parallelograms.
• When a semicircle with fixed diameter is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a sphere.
• The axis of the sphere is the straight line which remains fixed and about which the semicircle is turned.
• The center of the sphere is the same as that of the semicircle.
• A diameter of the sphere is any straight line drawn through the center and terminated in both directions by the surface of the sphere.
• When a right triangle with one side of those about the right angle remains fixed is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cone. And, if the straight line which remains fixed equals the remaining side about the right angle which is carried round, the cone will be right-angled; if less, obtuse-angled; and if greater, acute-angled.
• The axis of the cone is the straight line which remains fixed and about which the triangle is turned.
• And the base is the circle described by the straight in which is carried round.
• When a rectangular parallelogram with one side of those about the right angle remains fixed is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cylinder.
• The axis of the cylinder is the straight line which remains fixed and about which the parallelogram is turned.
• And the bases are the circles described by the two sides opposite to one another which are carried round.
• Similar cones and cylinders are those in which the axes and the diameters of the bases are proportional.
• A cube is a solid figure contained by six equal squares.
• An octahedron is a solid figure contained by eight equal and equilateral triangles.
• An icosahedron is a solid figure contained by twenty equal and equilateral triangles.
• A dodecahedron is a solid figure contained by twelve equal, equilateral and equiangular pentagons.

## Conic Section

In geometry, conic section, also called conics, is any curve produced by the intersection of a plane and a right circular cone. Depending on the angle of the plane relative to the cone, the intersection is a circle, an ellipse, a hyperbola, or a parabola.

1. Circle : If the cone is sliced parallel to the base, the resulting curve is a circle.
2. Ellipse: If the cone is sliced on a slight angle, the curve is called an ellipse.
3. Parabola : If the slice is made parallel to the edge of the cone, the curve formed is called a parabola.
4. Hyperbola: If the slice is perpendicular to the base of the cone, the curve is one of two branches of a hyperbola.

## Proportion

Proportion is the expression of equality between two equal ratios. Where ratio is defined as the comparison or simplified form of two quantities of the same kind. This relation indicates how many times one quantity is equal to the other; or in other words, ratio is a number which expresses one quantity as a fraction of the other. A ratio is an indication of the relative size of two magnitudes.

Symbols: A proportion is written in one of the following forms: a : b = c : d; a : b :: c : d; a/b = c/d. This proportion is read ” a is to b as c is to d “; or ” the ratio of a to b is equal to the ratio of c to d.”

Terms: In a proportion the four quantities compared are called the terms. The first and third terms are called the antecedents; the second and fourth terms, the consequents.

The first and fourth terms are called the extremes; the second and third terms, the means. Thus in the proportion a : b :: c : d, a and c are the antecedents, b and d the consequents, a and d the extremes, b and c the means.

Fourth Proportion: The fourth term of a proportion is called the fourth proportional to the terms taken in order. Thus in the proportion a:b = c: d, d is the fourth proportional to a, b, and c.

Continued Proportion: The quantities a, b, c, d, e are said to be in continued proportion, if a : b = b : c = c : d =.d : e. If three quantities are in continued proportion, the second is called the mean proportional between the other two, and the third is called the third proportional to the other two. Thus in the proportion a: b = b: c, b is the mean proportional between a and c, and c is the third proportional to a and b.

Theory of proportion – Proportion applied to similar figures

Some euclidean definitions:

1. Similar rectilinear figures are such as have their angles severally equal and the sides about the equal angles proportional.
2. Two figures are reciprocally related when the sides about corresponding angles are reciprocally proportional.
3. A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.
4. The height of any figure is the perpendicular drawn from the vertex to the base.