ClipArt ETC
A rolled out, or development, of the cylinder. The development is created by drawing the top curve with the dashed line where the folds are.

Development of Cylinder

A rolled out, or development, of the cylinder. The development is created by drawing the top curve with…

A development, or rolled out, cone where the point A is the meeting point for the sides. The development is drawn by drawing the arc by using a compass.

Development of Cone

A development, or rolled out, cone where the point A is the meeting point for the sides. The development…

A development, or rolled out image, of two cylinders intersecting each other. The large rectangular diagram is the main cylindrical body with a circle inside it for the other cylinder. The smaller development is the intersected cylinder. This is commonly used to illustrate pipes.

Development of Two Intersecting Cylinder

A development, or rolled out image, of two cylinders intersecting each other. The large rectangular…

This image represents an isometric holohedral mirror positioned to produce a cube by reflection.

Isometric Holohedral Mirror

This image represents an isometric holohedral mirror positioned to produce a cube by reflection.

This image shows one of Friedrich Froebel's divided cube (this one divided into eight smaller cubes). Froebel's cubes were used to encourage creativity in kindergarten-age children. The children could rearrange the smaller cubes into combinations that showed life, knowledge, and beauty.

Froebel's Divided Cube (Eight Smaller Cubes)

This image shows one of Friedrich Froebel's divided cube (this one divided into eight smaller cubes).…

This image shows one of Friedrich Froebel's divided cube (this one divided into eight smaller parallelograms). Froebel's cubes were used to encourage creativity in kindergarten-age children. The children could rearrange the smaller shapes into combinations that showed life, knowledge, and beauty.

Froebel's Divided Cube (Eight Smaller Parallelograms)

This image shows one of Friedrich Froebel's divided cube (this one divided into eight smaller parallelograms).…

This image shows one of Friedrich Froebel's divided cube (this one divided into twenty-seven smaller cubes). Froebel's cubes were used to encourage creativity in kindergarten-age children. The children could rearrange the smaller cubes into combinations that showed life, knowledge, and beauty.

Froebel's Divided Cube (Twenty-seven Smaller Cubes)

This image shows one of Friedrich Froebel's divided cube (this one divided into twenty-seven smaller…

This image shows one of Friedrich Froebel's divided cube (this one divided into many smaller cubes and prisms). Froebel's cubes were used to encourage creativity in kindergarten-age children. The children could rearrange the smaller shapes into combinations that showed life, knowledge, and beauty.

Froebel's Divided Cube (Complex)

This image shows one of Friedrich Froebel's divided cube (this one divided into many smaller cubes and…

Geometrical perspective drawing.

Geometry

Geometrical perspective drawing.

Geometrical plane.

Geometrical Plane

Geometrical plane.

Showing distance between two points on a coordinate plane.

Distance

Showing distance between two points on a coordinate plane.

A right triangle on a coordinate plane.

Right Triangle

A right triangle on a coordinate plane.

Showing how to find points on a coordinate plane. Also, shows an example of the midpoint formula.

Point Ratio

Showing how to find points on a coordinate plane. Also, shows an example of the midpoint formula.

An example of a quadrilateral used to find its area.

Area of Quadrilaterals

An example of a quadrilateral used to find its area.

A scalene triangle.

Scalene Triangle

A scalene triangle.

A perpendicular line on a coordinate plane, showing what y the y-coordinate will be.

Perpendicular Line

A perpendicular line on a coordinate plane, showing what y the y-coordinate will be.

An x=y line on a coordinate plane.

Straight Line

An x=y line on a coordinate plane.

To find the equation of a circle whose center is the origin and whose radius is 3 units in length.

Circle Equation

To find the equation of a circle whose center is the origin and whose radius is 3 units in length.

Polar coordinate example. Ex. (4, 60 degrees)

Coordinate Example

Polar coordinate example. Ex. (4, 60 degrees)

Changing from polar to rectangular coordinates.

Rectangular Coordinates

Changing from polar to rectangular coordinates.

Finding the distance between two points using polar coordinates.

Distance Between

Finding the distance between two points using polar coordinates.

Finding the area of a triangle in terms of polar coordinates of its three vertices.

Triangle Area

Finding the area of a triangle in terms of polar coordinates of its three vertices.

Equation of a straight line: a) parallel to y-axis, b) parallel to x-axis.

Straight Line

Equation of a straight line: a) parallel to y-axis, b) parallel to x-axis.

A straight line through the origin of a coordinate plane.

The Origin

A straight line through the origin of a coordinate plane.

A positive slope line making a 60 degree angle with the x-axis.

Straight Line

A positive slope line making a 60 degree angle with the x-axis.

A line with a standard form equation.

General Form

A line with a standard form equation.

A line with intercept form equation.

Intercept Form

A line with intercept form equation.

A line on the coordinate plane showing evidence of "perpendicular" form.

Perpendicular Form

A line on the coordinate plane showing evidence of "perpendicular" form.

Another method for using perpendicular form.

Alternate Method

Another method for using perpendicular form.

Other forms of line equations.

Line Equations

Other forms of line equations.

Finding the equation of a line drawn through a given point at a given direction.

Given Point

Finding the equation of a line drawn through a given point at a given direction.

Finding the equation of a line through two given points.

Line Equation

Finding the equation of a line through two given points.

Equation of a line with two given points and a given inclination.

Given Incline

Equation of a line with two given points and a given inclination.

Finding the angle between two lines whose equations are in intercept form.

Angle

Finding the angle between two lines whose equations are in intercept form.

Finding the perpendicular distance of points whose equation is x cosa + y sina = p.

Perpendicular Distance

Finding the perpendicular distance of points whose equation is x cosa + y sina = p.

An alternate form of the perpendicular distance method.

Alternate Form, Perpendicular Distance

An alternate form of the perpendicular distance method.

Multiple lines showing many bisectors on the coordinate plane.

Line Bisectors

Multiple lines showing many bisectors on the coordinate plane.

Any system of coordinate axes to another set which is parallel to the former, but with different origins.

Axes Shift

Any system of coordinate axes to another set which is parallel to the former, but with different origins.

Transforming from one set of rectangular axes to another with the same origin, but different direction.

Rectangular Axes

Transforming from one set of rectangular axes to another with the same origin, but different direction.

Finding the equation of a circle with the origin as its center.

Circle Equation

Finding the equation of a circle with the origin as its center.

A circle in its common, or central form. This is used to assist students in finding the equation of any circle.

Common Form of Circle

A circle in its common, or central form. This is used to assist students in finding the equation of…

A circle with many tangent and secant lines passing through it.

Lines

A circle with many tangent and secant lines passing through it.

Finding the equation of the normal at any point on a circle.

Normal Equation

Finding the equation of the normal at any point on a circle.

Tangent lines are drawn from a given external point to a circle. The equation is found by joining their points of contact

Chord Equation

Tangent lines are drawn from a given external point to a circle. The equation is found by joining their…

The chord of a circle.

Circle Chord

The chord of a circle.

Example of an equation with a given external point.

Equation Example

Example of an equation with a given external point.

If the chords of a circle are drawn through a fixed point, then the points of intersection of the pairs of tangents at the extremities of the chords will all lie on a fixed straight line.

Polar Property

If the chords of a circle are drawn through a fixed point, then the points of intersection of the pairs…

Polar property example.

Example

Polar property example.

Constructing the polar of a given point P with respect to a circle.

Polar of Point

Constructing the polar of a given point P with respect to a circle.

Finding the length of a tangent from a given point to a circle.

Tangent Length

Finding the length of a tangent from a given point to a circle.

Finding the polar equation of a straight line.

Polar Equation

Finding the polar equation of a straight line.

Finding the polar equation of a straight line by passing through two given points.

Straight Line

Finding the polar equation of a straight line by passing through two given points.

Finding the equation of a line with polar coordinates.

Polar Coordinates

Finding the equation of a line with polar coordinates.

An inscribed triangle in a circle.

Particular Cases

An inscribed triangle in a circle.

A circle with part of a triangle inscribed in it. They are lying on a coordinate plane.

Coordinate Plane

A circle with part of a triangle inscribed in it. They are lying on a coordinate plane.

A circle and triangle situated on coordinate planes.

Circle

A circle and triangle situated on coordinate planes.

Angles on a modified coordinate plane.

Oblique Axes

Angles on a modified coordinate plane.

A secant is "a line which cuts a figure in any way. Specifically, in trigonometry, a line from the center of a circle through one extremity of an arc (whose secant it is said to be) to the tangent from the other extremity of the same arc; or the ratio of this line to the radius; the reciprocal of the cosine. The ratio of AB to AD is the secant of the angle A; and AB is the secant of the arc CD." —Whitney, 1889

Circle with Secant

A secant is "a line which cuts a figure in any way. Specifically, in trigonometry, a line from the center…

"The aqueous vapor of the atmosphere precipitated in a crystalline form, and falling to the earth in flakes, each flake consisting of a distinct crystal, or more commonly combinations of separate crystals. The crystals belong to the hexagonal system, and are generally in the form of thin plates and long needles or spiculae; by their different modes of union they present uncounted varieties of very beautiful figures." —Whitney, 1889

Snowflakes as Described by William Scoresby

"The aqueous vapor of the atmosphere precipitated in a crystalline form, and falling to the earth in…

Illustration of a spiral named after the 3rd century BC Greek mathematician Archimedes.

Archimedean Spiral

Illustration of a spiral named after the 3rd century BC Greek mathematician Archimedes.