# 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

Angle | Notation

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

Cartesian Coordinate System | Notation

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.

Circle | Notation

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 | Notation

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

Congruent Angles | Notation

Congruent angles are denoted with tick marks across the angle.

Line | Notation

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

Parallel Lines | Notation

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 | Notation

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 | Notation

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

Polar Coordinate System | Notation

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 | Notation

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 | Notation

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

Theta | Notation

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

Triangle | Notation

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

## Formulas

Area of Circle | Formula

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

Area of Parallelogram | Formula

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

Area of Rectangle | Formula

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

Area of Trapezoid | Formula

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 | Formula

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

Circumference of Circle | Formula

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

Distance Between Two Points 1D | Formula

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 | Formula

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 | Formula

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.

Volume of Cone | Formula

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 | Formula

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

Volume of Cylinder | Formula

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

Volume of Rectangular Prism | Formula

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

Volume of Sphere | Formula

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

## Concepts

Acute Angle | Concept

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

Angle | Concept

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 | Concept

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

Cartesian Coordinate System | Concept

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 Extrema | Concept

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

Circle | Concept

Compass and Straight Edge Construction | Concept

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

Degrees | Concept

Degrees is a unit of measure for angles. A full rotation is equal to 360 degrees. In the cartesian coordinate system, degrees are measured starting from the rightmost edge of the circle.

Golden Ratio | Concept

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

Golden Rectangle | Concept

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

Inscribed Angle Theorem | Concept

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

Interval | Concept

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

Obtuse Angle | Concept

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

Perpendicular Angle | Concept

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

Perpendicular Lines | Concept

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

Pi | Concept

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

Point | Concept

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.

Polar Coordinate System | Concept

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

Pythagorean Theorem | Concept

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

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.

Slope | Concept

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

Sphere | Concept

Tau | Concept

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

Triangle | Concept

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.

Trigonometric Functions | Concept

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 | Concept

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

## Examples

A Plus B Squared | Example

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

Area of Circle Given Radius | Example

To calculate the area of a circle given the radius, multiply one half by the circle constant mutliplied by the radius squared.

Area of Rectangle | Example

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

Bisect Angle With Compass and Straight Edge | Example

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

Bisect Line With Compass and Straight Edge | Example

How to bisect line with compass and straight edge.

Circumference of Circle Given Radius | Example

To calculate the circumference of a circle multiply the radius of the circle by the circle constant tau approximately equal to 6.283.

Circumference of Circle Radius 1 | Example

To calculate the circumference of a circle multiply the radius by the constant τ (tau) to get the result.

Circumference of Circle Radius 2 | Example

To calculate the circumference of a circle multiply the radius by the constant τ (tau) to get the result.

Construct Ellipse With Two Tacks, String, and Pencil | Example

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

Construct Equilateral Triangle | Example

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 | Example

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

Construct Hexagon | Example

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 | Example

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 | Example

How to construct a perpendicular line given a straight line.

Derive Difference of Two Angles Identities | Example

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 Double Angles Identities (Algebra) | Example

The double angle identities can be derived using the sum of two angle identities in combination with Pythagorean's identity.

Derive Double Angles Identities (Inscribed Angle) | Example

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

Derive Half Angle Identities (Algebra) | Example

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 Law of Sines | Example

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) | Example

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 | Example

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.

Derive Sum of Two Angles Addition Formula | Example

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 | Example

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.

Distance Between 3 and 7 | Example

To find the distance between two points in one dimension, substitue the x values of the points into the formula for the distance between two points.

Distance Between 5 and 2 | Example

To find the distance between two points in one dimension, substitue the x values of the points into the formula for the distance between two points.

Line Through Two Points | Example

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

Midpoint of Line | Example

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.

Pythagorean Theorem 3 4 5 Right Triangle | Example

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

Volume of Cone Given Radius 1 Height 3 | Example

To calculate the volume of the cone with a radius of 1 and height of 3 substitute the values into the volume of a cone formula.

Volume of Cone Given Radius 2 Height 3 | Example

To calculate the volume of the cone with a radius of 2 and height of 3 substitute the values into the value of a cone formula.

Volume of Sphere Given Radius of 1 | Example

To calculate the volume of the cone with a radius of 1 and height of 3.

Volume of Sphere Given Radius of 2 | Example

To calculate the volume of the cone with a radius of 1 and height of 3.

## Interactives

Unit Circle | Interactive

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

Cosine Function | Interactive

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 | Interactive

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 | Interactive

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