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Fundamentals of Geometry

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Area Between Two Curves
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits at Infinity and Asymptotes
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Radius of Convergence
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Separation of Variables
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Conservation of Mechanical Energy
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Elastic Strings and Springs
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Power
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Work Done by a Constant Force
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Argand Diagram
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- De Moivre's Theorem
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Distribution Function
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Degrees of Freedom
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test for Regression Slope
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Paired T-Test
- Point Estimation
- Probability
- Probability Calculations
- Probability Density Function
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residual Sum of Squares
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Skewness
- Spearman's Rank Correlation Coefficient
- Standard Deviation
- Standard Error
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- T-distribution
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Variance for Binomial Distribution
- Venn Diagrams

Though it can seem complicated at times, **geometry** can essentially boil down to a few **fundamental concepts**. These concepts have been known for thousands of years, with their origins in several different ancient cultures. The ancient Greek mathematician **Euclid** is often considered the '**father' of geometry**, with his **f****undamentals of Euclidean geometry** serving as a basis for much of 'modern mathematics' understanding of **geometry**.

In uncovering the secrets of the fundamentals of geometry, Euclid realised that first must come the very basic fundamental concepts. Certain questions had to be answered to define these fundamental constituents of the field, such as *what is a point, *or *what is a line?*

Euclid defined a point in the following way: “a point is that which has no part." In essence, this simply means that a point is just a location in space that has no dimensions. That is to say, though it has spatial parameters to define its location, **it does not actually occupy any space itself**.

Euclid also defined something called a **line**. This he defined as "a length without a breadth." In essence, just a 1-dimensional segment with a finite length. He posited that a line could be extended infinitely in either direction. This is an area where modern geometry differs from Euclid's fundamentals, as now we refer to an infinitely extended line simply as a **line** and Euclid's line of finite length as a **line segment**. This is an important distinction to remember for the sake of correctness. Let's take a look at the difference below. Euclid also defined something called a *ray*, which similar to a line is infinitely long, however, it has a defined starting point. It is also sometimes known as a *half-line.*

A line is a straight, 1-dimensional figure that extends indefinitely in both directions. A line segment is a straight, 1-dimensional figure of finite length that connects two points. A ray is a straight one-dimensional figure that extends infinitely in one direction, from a defined starting point.

A plane, in many ways, can be considered as similar to a line, but in two dimensions. A plane is simply a surface that extends indefinitely. A plane can exist in 2-dimensional spaces as well as 3-dimensional spaces and higher.

A plane is a 2-dimensional figure that extends indefinitely in four directions.

Angles were defined by Euclid as "the inclination of two straight lines." This essentially can be described in simpler terms as the rotational distance between two lines or line segments that share a point, i.e how much would we have to rotate one line before it lines up with the other. The shared point is known as the **vertex **of the angle.

An angle is a measure of rotational space between two lines or line segments.

Dimensions are an important aspect of the fundamentals of geometry, which specifically deals with **spatial dimensions**. Spatial dimensions in mathematics and physics can be defined as the minimum number of coordinates required to describe a point in that space. For instance, a line has 1-dimensions as only a single number is required to specify a point on that line. Equally, if you wanted to specify a point on an x-y axis you would need two coordinates an x and y coordinate and on a set of 3-dimensional axes, you would need a third coordinate - the z coordinate.

Dimensions are extensions of space in a single direction, the length along which can be used to describe the location of a point in that dimension. Multiple dimensions can be combined to describe geometric properties with increasing complexity.

Area is a measurement that describes the size of a certain 2-dimensional region. There are various formulas that can be used in calculating the area of certain shapes. A good way to visualise area is to divide up a 2-dimensional space into squares. The area of the shape is simply equal to the number of squares contained inside it.

Area is a measurement that describes the size of a certain 2-dimensional region of space.

Much like area, volume is a measure that quantifies the size of a certain region of space. Volume, however, quantifies the size of a region in **3-dimensional **space. All 3-dimensional shapes have volume, and similar to area there are many useful formulas for calculating the volumes of various shapes. We can visualise volume in much the same way as area, but rather than using small squares, we count the number of small cubes inside a shape. The image below depicts a cube in 3-dimensional space. How much space does the cube take up? Well, by counting we can see that the cube takes up the space of 64 smaller cubes, each with a volume of 1 units^{3}.

Volume is a measurement that describes the size of a certain 3-dimensional region of space.

A cube in 3-dimensional space,

John Hannah - StudySmarter Originals

An important part of the fundamentals of geometry is the use of various units. In geometry, we use two basic types of units: units of **length** and units of **angles****.**

A unit is a convention that helps us define how large something is. For instance, a unit of **length** can help us define how long something is, and a unit of volume can help us define how large a 3-dimensional shape is.

There are two primary systems for units of length. These systems are the **metric **and **imperial **systems. The metric system deals in units of centimetres, metres, kilometres etc. whereas the imperial system works in units inches, feet, yards, miles etc.

Length is a 1-dimensional unit, however, units of 2-dimensions (area) and 3-dimensions (volume) exist which are composed of these units of length. The convention for the naming of these dimensions is shown in the table below.

Length | Area | Volume |

Degrees and radians are the two main units for the measurement of angles and it's very easy to run into problems if the distinction between the two isn't clear!

Firstly, it is important to recognise that degrees are an arbitrary unit of measurement that simply arose from the fact of the Earth's rotation. Ancient people watching the constellations in the sky move on a yearly cycle figured that since there were 360 days in a year (there are really 365 but they got pretty close!) that there should be 360 degrees in a full rotation. This has proved a simple and intuitive way to discuss angles as human beings, after all, we aren't computers and sometimes closely packed decimal numbers can be confusing.

However, since these early scholars sat looking at the stars, we have discovered another, arguably more mathematically sound way of describing angles. This unit is known as the **radian**.

Radians, rather than being related simply to 'amount of rotation', are related to distance travelled around an arc. Really, radians are in fact **the distance travelled around an arc divided by the distance to the pivot point of that arc**. If we take the equation relating a circle's circumference to its radius, we can find how many radians in a full rotation of 360^{o}.

So in 360^{o} there are radians. From this, we can see that

Degrees | Radians |

360^{o} | |

180^{o} | |

90^{o} | |

45^{o} |

It is important when using a calculator if it is set to treat your angles as **radians **or **degrees** when dealing with trigonometric functions to obtain the correct answer. Technically all mathematical functions taking angles as inputs work in radians.

Euclid made five fundamental postulates when delving into the field of geometry. These postulates were fundamental principles of geometry that he held to be self-evident, and informed all further principles and concepts of geometry thereafter.

Postulates of Euclidean Geometry |

1. A straight line segment can be drawn joining any two points. |

2. Any straight line segment can be extended indefinitely in a straight line. |

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre. |

4. All right angles are congruent. |

5. Given a line and a point not on that line, there exist an infinite number of lines through the given point parallel to the given line. |

- Many of the fundamentals of geometry that are used today were popularised by a Greek mathematician known as Euclid.
- A point is a single location space that has no size.
- A line is a 1-dimensional figure that extends indefinitely.
- A line segment is a 1-dimensional figure with finite start and end-points.
- A plane is a 2-dimensional figure that extends indefinitely.
- Angles are a measure of rotational distance.
- Dimensions can be described as the coordinates required to define a point in a certain space.
- Area is a measure of the 2-dimensional size of a shape.
- Volume is the measure of the 3-dimensional size of a shape.
- Units are a convention used to define the size of various quantities.
- Units of length are either of the imperial system or the metric system.
- Units of angles are either radians or degrees.
- Euclid defined five postulates of geometry that he held as self-evident truths.

The most basic, fundamental components of geometry are points, lines, and planes.

*a line segment splits two sides of a triangle into proportional segments if and only if the segment is parallel to the triangle's third side.*

More about Fundamentals of Geometry

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