Geometry Index

Geometry

Geometry is the study of space and shape. Modern mathematics approaches geometry from the view point of the Cartesian Coordinate System. The Cartesian Coordinate System has an origin and the position of of a point in space is measured by its distance from this origin.

Notation

Slope

Slope is denoted as the change in y over the change in x. The capital greek letter delta (Δ) is used to represent the change in a variable.

Cartesian Coordinate System

A point in the cartesian coordinate system is denoted by two numbers in parentheses separated by a comma. The first number represents the distance from origin in the x-direction and teh second number represents the distance from the origin in the y-direction.

Polar Coordinate System

A point in the polar coordinate system is denoted by two numbers in parentheses separated by a comma. The first number represents the radius r (distance from the origin) and an angle θ (the greek letter theta) relative to the origin.

Angle

The notation for a symbol is a small symbol written in text, sometimes followed by three letters that correspond to a figure.

Circle

A circle in text is denoted using the circle symbol with a dot in the center followed by a letter that corresponds to the center point of the circle.

Complementary Angles

Complementary angles can visually be denoted as two angles who sum to a perpendicular or square angle.

Congruent Angles

Congruent angles are denoted with tick marks across the angle.

Line

A line is denoted by two letters representing the start and end of the line with a line over top.

Parallel Lines

Parallel lines are denoted by the parallel symbol placed betwen the notation of the two lines. A line is denoted by the start and end letter with a line over top.

Perpendicular Angle

A perpendicular angle is visually denoted by drawing a square at the vertex of the angle. The measured angle is equal to π/2 radians or 90°.

Perpendicular Lines

The symbol for two perpendicular lines is a horizontal line with another line drawn perpendicular to it.

Supplementary Angles

Supplementary angles can visually be denoted as two angles who sum to 180 degrees or PI degrees.

Triangle

A triangle is denoted using the triangle symbol followed by three letters that represent the points of the triangle.

Formulas

Area of Circle

The area of a circle is give by one-half multiplied by τ (tau) mutliplied by the radius of the circle squared.

Area of Circle π (pi)

The traditional formula for the area of a circle is given in terms of the geometric constant π (pi).

Area of Parallelogram

The area of any parallelogram is equal to the base multiplied by the height.

Area of Rectangle

The Area of a rectangle is given by its width multiplied by its height.

Area of Trapezoid

The area of a trapezoid is given by its height multiplied by the sum of its top length and bottom length divided by two.

Area of Triangle

The area of a triangle is given by one half multiplied by its width and height.

Circumference of Circle

The circumference of a circle is given the constant τ (tau) multpilied by the radius of the circle, where τ = 2π.

Circumference of Circle π (pi)

The circumference of a circle is given the constant π (pi) multpilied by two times the radius of the circle.

Distance Between Two Points 1D

The distance between two points, in one dimension, is given by the absolute value of the difference between the two values.

Distance Between Two Points 2D

The distance between two points, in two dimensions, is given by solving pythagorean's theorem for the length the hypotenuse of the right triangle formed by the two points.

Midpoint Formula

The x-coordinate of the mid-point is calculated by averaging the x-coordinates of the two end points. The y-coordinate is calculated by averaging the y-coordinates of the two points.

Slope

The slope of a line is calculated by finding the change in y over the change in x using two points on the line.

Volume of Cone

The volume of a cone is given by one third multiplied by PI, the radius of its base squared, and its height.

Volume of Cube

The volume of a cube is given by the length of the cube length raised to the third power.

Volume of Cylinder

The volume of a cylinder is given by 1/2 τ (tau) multiplied by the radius squared and height.

Volume of Rectangular Prism

To calculate the volume of a rectangular prism multiply its height, width, and length together.

Volume of Sphere

The volume of a sphere is given by two-thirds multiplied by the circle constant τ (tau) multiplied by the radius cubed.

Concepts

Radian Angle System

Radians are a unit that measure angle using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle.

Cartesian Coordinate System

The Cartesian Coordinate System describes space of one, two, and three dimensions. Each point in space is represented by its distance relative to the origin of the system.

Compass and Straight Edge Construction

Geometric construction is a classic form of math that studies building forms and shapes using a compass and straight edge.

Degree Angle System

Degrees are a unit of measure for angles. A full rotation is equal to 360 degrees. In the XY Cartesian Coordinate System, degrees are measured starting from the rightmost edge of the circle.

Golden Ratio

The golden ratio is a number represented by the greek letter ϕ (phi). The value of ϕ is approximately 1.618 and is a naturally occurring number in nature.

Golden Rectangle

The golden rectangle is a rectangle whose width divided by height is equal to the golden number (approximately 1.618).

Inscribed Angle Theorem

The inscribed angle theorem states that an inscribed angle in a circle is equal to one-half the central angle.

Pythagorean Theorem

The pythagorean theorem equates the square of the sides of a right triangle together.

Trigonometric Functions

There are six trigonometric functions that relate to the geometry of the right-triangle sine, cosine, tangent, cosecant, secant, and cotangent. The functions take the angle of a right triangle as input and return a ratio of two of its sides.

Trigonometric Functions Introduction

There are six trigonometric functions that relate to the geometry of the right-triangle sine, cosine, tangent, cosecant, secant, and cotangent. The functions take the angle of a right triangle as input and return a ratio of two of its sides.

τ (tau)

The circle constant τ (tau) is a geometric constant approximately equal to 6.283. The numeric value is defined as the length of any circle's circumference divided by the length of its radius.

Examples

Bisect Angle

How to divide an angle into two equal parts using a compass and straight edge.

Bisect Line

This exampls shows how to bisect a line with compass and straight edge.

Calculate A Plus B Squared

This example calculates the result of a plus b squared using the area of the corresponding square.

Calculate Line Through Two Points

To find the equation of a line given two points first calculate the slope of the line and then the y-intercept.

Construct Equilateral Triangle

This example demonstrates how to construct an equilateral triangle using a compass and ruler. An equilateral triangle is a triangle whose sides are of equal length.

Construct Golden Rectangle

How to build the golden rectangle using a compass and ruler. This geometric construction also demonstrates how to find the golden ratio.

Construct Hexagon

This example demonstrates how to construct an hexagon using a straight edge and ruler. A hexagon is a polygon with six equal length sides.

Construct Pentagon

This example demonstrates how to construct an pentagon using a straight edge and ruler. A hexagon is a polygon with five equal length sides.

Construct Perpendicular Line

How to construct a perpendicular line given a straight line.

Derive Difference of Two Angles Identities

To derive the difference of two angles identities, two right triangles are placed next to eachother so their angles sum together to be one angle and one triangle's angle is the difference of the sum and the other.

Derive Difference of Two Angles Identities (Complex Plane)

This example demonstrates who to derive the difference of two angle identities.

Derive Double Angle Identities (Algebra)

This example derives the double angle identities using algebra and the sum of two angles identities.

Derive Double Angles Identities (Complex Plane)

The trigonometric double angle identites can be derived using the properties of the complex plane by defining a complex number in terms of the sine and cosine of its angle.

Derive Double Angles Identities (Inscribed Angle)

The double angle identities can be derived using the inscribed angle theorem on the circle of radius one.

Derive Half Angle Identities (Algebra)

The half-angle identities can be derived from the double angle identities by transforming the angles using algebra and then solving for the half-angle expression.

Derive Half Angle Identities (Complex Plane)

This example derives the trig. half angle identities in the complex plane.

Derive Half Angle Identities (Inscribed Angle)

This example demosntrates how to derive the trig. half angle identities using the inscribed angle theorem.

Derive Law of Sines

To derive the equation for the law of sines, observe the shared perpendicular line by two angles. The third angle can be included by repeating the process.

Derive Law of Sines (Inscribed Triangle)

The law of sines can be derived using a triangle inscribed on the perimeter of a circle. The proof uses the inscribed angle theorem.

Derive Point Where Two Lines Intersect

This example demonstrates the general process for finding the point where two lines intersect, where the equations of the lines are given in the slope intercept form.

Derive Sum of Two Angles (Ptolemy's Theorem)

The sum of two angles addition formula can be derived using a quadrilateral inscribed on a circle of diameter 1.

Derive Sum of Two Angles (Unit Circle)

The sum of two angles addition formula can be derived using a quadrilateral inscribed on a circle of diameter 1.

Derive Sum of Two Angles Identities

To derive the sum of two angles identities, two right triangles are placed next to eachother so their angles sum together, then their proportions are related together.

Derive Sum of Two Angles Identities (Complex Plane)

The sum of two angles identities can be derived using the properties of the complex plane by defining a complex number in terms of the sine and cosine of its angle.

Interactives

Degrees

This interactive animates the degree angle system where a full rotation is equal to 360 degees. The angle is shown in standard position.

Unit Circle

This interactive demonstrates the connection between the unit circle and the trigonometric functions sine, cosine and tangent.

Cosine Function

This interactive demonstrates the connection between a right triangle of hypotenuse one and the graph of the cosine function. Click and drag either point to change the angle that describes the right triangle, or change the angle in the input box.

Sine Function

This interactive demonstrates the connection between every right triangle of hypotenuse one and the graph of the sine function. Click and drag either point to change the angle that describes the right triangle, or change the angle in the input box.

Angle

This interactive illustrates how an angle is defined by two rays. The point where the two meet is called the vertex of the angle.