# Geometry Index

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

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

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.

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

Congruent angles are denoted with tick marks across the angle.

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

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.

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°.

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

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.

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.

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

The symbol theta is often used as a variable to represent an angle in illustrations, functions, and equations.

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

## Formulas

The area of a circle is given by π (pi) times the radius of the circle squared.

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

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

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

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

The circumference of a circle is given by two multiplied by pi and the radius.

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

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.

To find the midpoint between two points, average the two points x coordinate together to get the midpoint's x coordinate, then average the two points y coordinate together to get the midpoints y coordinate.

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

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

The volume of a cylinder is equal to PI multiplied by its radius squared and its height.

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

The volume of a sphere is equal to four over three multiplied by PI and its radius cubed.

## Concepts

An acute angle is an angle that is smaller than 90 degrees or PI fourths.

An angle is defined as the amount of rotation between two rays. Angles are measured using degrees and radians. A full rotation in degrees is 360°. A full rotation in radians is approximately 6.283 radians or τ (tau) radians.

Area is the physical amount of two dimensional space that a shape takes up. Area is measured in square units.

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.

Chebyshev are the roots to a special series of polynomial equations.

This page describes the basic elements and properties of a circle.

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

Degrees are a unit that measures angle of rotation as a fraction of the number 360.

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 occuring number in nature. The golden ratio is often associated with the golden rectangle whose sides form a ratio equal to ϕ.

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

In math an interval represents the valid values between an upper limit and a lower limit.

An obtuse angle is an angle that is larger than 90 degrees or PI fourths.

A Perpendicular angle, sometimes also referred to as a square angle, is exactly 90 degrees or PI fourths.

A Perpendicular angle, sometimes also referred to as a square angle, is exactly 90 degrees or PI fourths.

The greek letter π (pi) is a naturally occuring number that is defined by any circle's circumference divided by its diameter.

A point represents a position in space. In modern mathematics, space is represented using the cartesian coordinate system where there is an origin and the position of a point in space is measured by its distance from the origin.

The Polar Coordinate System describes points in space using an angle and radius relative to the origin.

The pythagorean theorem is an equation that equates the square of the sides of a right triangle together.

Radians are a unit that measure angles 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.

Slope is the concept of how much a function is changing with respect to x.

This page describes the basic elements and properties of a sphere.

τ (tau) is a geometric constant approximately equal to 6.283. The number is naturally occuring as any circle's circumference divided by its radius.

A triangle is a three sided geometric shape. The shape forms a basis for the subject of trigonometry and is used throughout mathematics and programming.

There are six total 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.

Volume measures the amount of three dimensional space an object occupies.

## Examples

This example dmemonstrates how the general form of a plus b squared can be interpretted geometrically.

To calculate pi geometrically we can use the perimeter of a polygon to approximate the circumference of a circle. This combined with the definition of pi, circumference divided by diameter, allows us to calculate a value for pi.

To calculate the area of a circle when given a radius you can use the formula for the area of the circle.

To find the area of a rectangle given its width and height, multiply the width and height together.

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

How to bisect line with compass and straight edge.

The Circumference of a circle is given by two multiplied by PI and the radius of the circle.

This page explains how to draw a bezier curve using a ruler.

This video shows how to construct an ellipse using two tacks, a string, and pencil.

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.

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

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

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

How to construct a perpendicular line given a straight line.

The cross product of two vectors can be visualized as the area of the parallelogram formed by the two vectors.

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.

How to find the distance between two points using pythagorean's theorem.

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.

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

To find the midpoint between two points, average the two points x coordinate together to get the midpoint's x coordinate, then average the two points y coordinate together to get the midpoints y coordinate.

To find the perimeter of a polygon with two to the n sides, we can use the recursive formula derived from the special triangle 45 45 90.

To find the point where two lines intersect set the equations equal together and solve for the x-coordinate. Then substitute the solved for coordinate back into one of the equations to get the y-coordinate.

This example shows how the pythagorean theorem is true for a right triangle with sides length 3, 4, and 5.

## Interactives

This page displays the connection between the unit circle and the trigonometric functions sine and cosine.

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.

This interactive demonstrates how the geometric constant Tau (τ) can be approximated using two polygons. One polygon is inscribed in the circle, the other is cricumscribed around the circle. Then using the perimeter of the two polygons a lower-bound and upper-bound approximation can be calculated. This is because τ is defined geometrically as the length of any circle's circumference dvided by the length of its radius.

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.

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

This interactive demonstrates a cubic Bézier curve described by four points.

This interactive demonstrates a quadratic Bézier curve described by three points.