ClipArt ETC
An illustration showing how to construct a tangent to 2 given circles of different diameters. "Join the centers C and c of the given circles, and extend the line to D; draw the radii AC and ac parallel with one another. Join Aa, and extend the line to D. On CD as a diameter, draw the half circle CeD; on cD as a diameter, draw the half circle cfD; then the crossings e and f are the tangenting points of the circles."

Construction Of Tangent To 2 Circles

An illustration showing how to construct a tangent to 2 given circles of different diameters. "Join…

An illustration showing how to construct a tangent between 2 given circles. "Join the centers C and c of the given circles; draw the dotted circle arcs, and join the crossing m, n, which line cuts the center line at a. With aC as diameter, draw the half circle afC; and with ac as a diameter, draw the half circle cea; then the crossings e and f are the tangenting points of the circles."

Construction Of Tangent Between 2 Circles

An illustration showing how to construct a tangent between 2 given circles. "Join the centers C and…

An illustration showing how to construct a circle tangent to a given line and given circle. "Add the given radius r to the radius R of the circle, and draw the arc cd. Draw the line ce parallel with and at a distance r from the line AB. Then the crossing c is the center of the required circle that will tangent the given line and circle."

Construction Of A Circle Tangent To A Line And A Circle

An illustration showing how to construct a circle tangent to a given line and given circle. "Add the…

An illustration showing how to construct the center and radius of a circle that will tangent a given circle. "Through the given point C, draw the tangent GF; bisect the angle FGE; then o is the center of the required circle that will tangent AB at C, and the line DE."

Construction Of A Center And Radius Of A Circle That Will Tangent A Given Circle

An illustration showing how to construct the center and radius of a circle that will tangent a given…

An illustration showing how to construct the center and radius of a circle that will tangent a given circle and line. "Through the given point C, draw the line EF at right angles to AB; set off from C the radius r of the given circle. Join G and F. With G and F as centers draw the arc crosses m and n. Join mn, and where it crosses the line EF is the center of the required circle."

Construction Of A Center And Radius Of A Circle That Will Tangent A Given Circle And Line

An illustration showing how to construct the center and radius of a circle that will tangent a given…

An illustration showing how to construct the center and radius of a circle that will tangent a given circle and line. "From C, erect the perpendicular CG; set off the given radius r from C to H. With H as a center and r as radius, draw the cross arcs on the circle. Through the cross arcs draw the line IG; then G is the center of the circle arc FIC, which tangents the line at C and the circle at F."

Construction Of A Center And Radius Of A Circle That Will Tangent A Given Circle And Line

An illustration showing how to construct the center and radius of a circle that will tangent a given…

An illustration showing how to construct two circles that tangent themselves and two given lines. "Draw the center line AB between the given lines; assume D to be the tangenting point of the circles; draw DC at right angles to AB. With C as center and CD as radius, draw the circle EDF. From E, draw Em at right angles to EF; and from F draw Fm at right angles to FE; then m and n are the centers for the required circles."

Construction Of Two Circles That Tangent Themselves and 2 Given Lines

An illustration showing how to construct two circles that tangent themselves and two given lines. "Draw…

An illustration showing how to construct a circle that tangents two given lines inclined to one another with the one tangenting point being given. "Draw the center line GF. From E, draw EF at right angles to AB; then F is the center of the circle required.

Construction Of A Circle That Tangents 2 Given Lines

An illustration showing how to construct a circle that tangents two given lines inclined to one another…

An illustration showing how to construct a circle that tangents two given lines and goes through a given point c on the line FC, which bisects the angle of the lines. "Through C draw AB at right angles to CF; bisect the angles DAB and EBA, and the crossing on CF is the center of the required circle."

Construction Of A Circle That Tangents 2 Given Lines And Goes Through A Given Point

An illustration showing how to construct a circle that tangents two given lines and goes through a given…

An illustration showing how to construct a cyma, or two circle arcs that will tangent themselves, and two parallel lines at given points A and B. "Join A and B; divide AB into four equal parts and erect perpendiculars. Draw Am at right angles from A, and Bn at right angles from B; then m and n are the centers of the circle arcs of the required cyma."

Construction Of A Cyma

An illustration showing how to construct a cyma, or two circle arcs that will tangent themselves, and…

An illustration showing how to construct a talon, or two circle arcs that will tangent themselves, and meet two parallel lines at right angles in the given points A and B. "Join A and B; divide AB into four equal parts erect perpendiculars; then m and n are the centers of the circle arcs of the required talon."

Construction Of A Talon

An illustration showing how to construct a talon, or two circle arcs that will tangent themselves, and…

An illustration showing how to construct a circle arc without recourse to its center, but its chord AB and height h being given. "With the chord as radius, and A and B as centers, draw the dotted circle arcs AC and BD. Through the point O draw the lines AOo and BOo. Make the arcs Co=Ao and Do=Bo. Divide these arcs into any desired number of equal parts, and number them as shown on the illustration. Join A and B with the divisions, and the crossings of equal numbers are points in the circle arc."

Construction Of A Circle Arc

An illustration showing how to construct a circle arc without recourse to its center, but its chord…

An illustration showing how to construct a center and radius of a circle that will tangent the three sides of a triangle. "Bisect two of the angles in the triangle, and the crossing C is the center of the required circle."

Construction Of The Center And Radius Of A Circle Tangent To Triangle Sides

An illustration showing how to construct a center and radius of a circle that will tangent the three…

An illustration showing how to construct an equilateral triangle inscribed in a circle. "With the radius of the circle and center C draw the arc DFE; with the same radius, and D and E as centers, set off the points A and B. Join A and B, B and C, C and A, which will be the required triangle."

Construction Of An Equilateral Triangle Inscribed In A Circle

An illustration showing how to construct an equilateral triangle inscribed in a circle. "With the radius…

An illustration showing how to construct a square inscribed in a circle. "Draw the diameter AB, and through the center erect the perpendicular CD, and complete the square as shown in the illustration."

Construction Of A Square Inscribed In A Circle

An illustration showing how to construct a square inscribed in a circle. "Draw the diameter AB, and…

An illustration showing how to construct a square circumscribed about a circle. "Draw the diameters AB and CD at right angles to one another; with the radius of the circle, and A, B, C, and D as centers, draw the four dotted half circles which cross one another in the corners of the square, and thus complete the problem."

Construction Of A Square Circumscribed About A Circle

An illustration showing how to construct a square circumscribed about a circle. "Draw the diameters…

An illustration showing how to construct a pentagon inscribed in a circle. "Draw the diameter AB, and from the center C erect the perpendicular CD. Bisect the radius AC at E; with E as center, and DE as radius, draw the arc DE, and the straight line DF is the length of the side of the pentagon."

Construction Of A Pentagon Inscribed In A Circle

An illustration showing how to construct a pentagon inscribed in a circle. "Draw the diameter AB, and…

An illustration showing how to construct a pentagon on a given line. "From B erect BC perpendicular to and half the length of AB; join A and C prolonged to D; with C as center and CB as radius, draw the arc BD; then the chord BB is the radius of the circle circumscribing the pentagon. With A and B as centers, and BD as radius, draw the cross O in the center."

Construction Of A Pentagon On A Line

An illustration showing how to construct a pentagon on a given line. "From B erect BC perpendicular…

An illustration showing how to construct a hexagon in a given circle. "The radius of the circle is equal to the side of the hexagon."

Construction Of A Hexagon In A Circle

An illustration showing how to construct a hexagon in a given circle. "The radius of the circle is equal…

An illustration showing how to construct a heptagon, or septagon. "The appotem a in a hexagon is the length of the side of the heptagon. Set off AB equal to the radius of the circle; draw a from the center C at right angles to AB; then a is the required side of the heptagon."

Construction Of A Heptagon

An illustration showing how to construct a heptagon, or septagon. "The appotem a in a hexagon is the…

An illustration showing how to construct an octagon on a given line. "Prolong AB to C. With B as center and AB as radius, draw the circle AFDEC; from B, draw BI at right angles to AB; divide the angles ABC and DBC each into two equal parts; then BD is one side of the octagon. With A and E as centers, draw the arcs HKE and AKI, which determine the points H and I, and thus complete the octagon as shown in the illustration."

Construction Of An Octagon

An illustration showing how to construct an octagon on a given line. "Prolong AB to C. With B as center…

An illustration showing how to construct a regular octagon from a square by cutting off the corners of the square. "With the corners as centers, draw circle arcs through the center of the square to the side, which determine the cut-off."

Construction Of An Octagon From a Square

An illustration showing how to construct a regular octagon from a square by cutting off the corners…

An illustration showing how to construct a regular polygon on a given line without resort to its center. "Extend AB to C and, with B as center, draw the half circle ADB. Divide the half circle into as many parts as the number of sides in the polygon, and complete the construction as shown on the illustration."

Construction Of A Regular Polygon On A Line

An illustration showing how to construct a regular polygon on a given line without resort to its center.…

An illustration showing how to construct an isometric ellipse by compass and six circle arcs. "Divide OA and OB each into three equal parts; draw the quadrant AC. From C, draw the line Cc through the point 1. Through the points 2 draw de at an angle of 45° with the major axis. Then 2 is the center for the ends of the ellipse; e is the center for the arc dc; and C is the center for the arc cf."

Construction Of An Isometric Ellipse

An illustration showing how to construct an isometric ellipse by compass and six circle arcs. "Divide…

An illustration showing how to construct an ellipse using circle arcs. "Divide the long axis into three equal parts, draw the two circles, and where they intersect one another are the centers for the tangent arcs of the ellipses as shown by the figure."

Construction Of An Ellipse

An illustration showing how to construct an ellipse using circle arcs. "Divide the long axis into three…

An illustration showing how to construct an ellipse using circle arcs. "Given the two axes, set off the short axis from A to b, divide b into three equal parts, set off two of these parts from o towards c and c which are the centers for the ends of the ellipse. Make equilateral triangles on cc, when ee will be the centers for the sides of the ellipse. If the long axis is more than twice the short one, this construction will not make a good ellipse."

Construction Of An Ellipse

An illustration showing how to construct an ellipse using circle arcs. "Given the two axes, set off…

An illustration showing how to construct an ellipse. Given the two axes, set off half the long axis from c to ff, which will be the two focuses in the ellipse. Divide the long axis into any number of parts, say a to be a division point. Take Aa as radius and f as center and describe a circle arc about b, take aB as radius and f as center describe another circle arc about b, then the intersection b is a point in the ellipse, and so the whole ellipse can be constructed."

Construction Of An Ellipse

An illustration showing how to construct an ellipse. Given the two axes, set off half the long axis…

An illustration showing how to construct an ellipse parallel to two parallel lines A and B. "Draw a semicircle on AB, draw ordinates in the circle at right angle to AB, the corresponding and equal ordinates for the ellipse to be drawn parallel to the lines, and thus the elliptic curve is obtained as shown by the figure."

Construction Of An Ellipse Tangent To Two Parallel Lines

An illustration showing how to construct an ellipse parallel to two parallel lines A and B. "Draw a…

An illustration showing how to construct a cycloid. "The circumference C=3.14D. Divide the rolling circle and base line C into a number of equal parts, draw through the division point the ordinates and abscissas, make aa' = 1d, bb' = 2'e, cc = 3f, then ab' and c' are points in the cycloid. In the Epicycloid and Hypocycloid the abscissas are circles and the ordinates are radii to one common center."

Construction Of A Cycloid

An illustration showing how to construct a cycloid. "The circumference C=3.14D. Divide the rolling circle…

An illustration showing how to construct an evolute of a circle. "Given the pitch p, the angle v, and radius r. Divide the angle v into a number of equal parts, draw the radii and tangents for each part, divide the pitch p into an equal number of equal parts, then the first tangent will be one part, second two parts, third three parts, etc., and so the Evolute is traced."

Construction Of An Evolute Of A Circle

An illustration showing how to construct an evolute of a circle. "Given the pitch p, the angle v, and…

An illustration showing how to construct a parabola. "Given the axis of ordinate B, and vertex A. Take A as a center and describe a semicircle from B which gives the focus of the parabola at f. Draw any ordinate y at right angle to the abscissa Ax, take a as radius and the focus f as a center, then intersect the ordinate y, by a circle-arc in P which will be a point in the parabola. In the same manner the whole Parabola is constructed."

Construction Of A Parabola

An illustration showing how to construct a parabola. "Given the axis of ordinate B, and vertex A. Take…

An illustration showing a circle with radius r, diameter d, and chord c.

Radius, Diameter, and Chord In A Circle

An illustration showing a circle with radius r, diameter d, and chord c.

An illustration showing a circle sector with radius r, center/central angle v, and length of circle arc b.

Circle Sector

An illustration showing a circle sector with radius r, center/central angle v, and length of circle…

An illustration showing a circle sector with center/central angle v and polygon angle w.

Circle Sector

An illustration showing a circle sector with center/central angle v and polygon angle w.

An illustration showing a circle sector with height of segment h and radius r.

Circle Sector

An illustration showing a circle sector with height of segment h and radius r.

An illustration showing a triangle with sides b, c, and d inscribed in a circle with radius r.

Triangle Inscribed In A Circle

An illustration showing a triangle with sides b, c, and d inscribed in a circle with radius r.

An illustration showing a circle with radius r inscribed in a triangle with sides a, b, and c.

Circle Inscribed In A Triangle

An illustration showing a circle with radius r inscribed in a triangle with sides a, b, and c.

An illustration showing a quadrilateral inscribed in a circle that is tangent to a line.

Quadrilateral Inscribed In A Circle

An illustration showing a quadrilateral inscribed in a circle that is tangent to a line.

An illustration showing a triangle with angles A, C, and D inscribed in a circle which is tangent to a line.

Triangle Inscribed In A Circle

An illustration showing a triangle with angles A, C, and D inscribed in a circle which is tangent to…

An illustration showing a circle with sectors D and E inscribed in a triangle with angles, A, B, and C.

Circle Inscribed In A Triangle

An illustration showing a circle with sectors D and E inscribed in a triangle with angles, A, B, and…

An illustration showing a model that illustrates the following relationships: a:x = x:a - x, x = √(a&sup2 + (a/2)&sup2 - a/2).

Model Of Geometric Proportions

An illustration showing a model that illustrates the following relationships: a:x = x:a - x, x = √(a²…

An illustration of a right triangle inscribed in a semicircle.

Right Triangle Inscribed In A Semicircle

An illustration of a right triangle inscribed in a semicircle.

An illustration showing a model of a circle with intersecting chords that illustrates the following relationship: a:c = b:d, ad = bc. Product of the means equals the product of the extremes.

Model Of Geometric Proportions In A Circle

An illustration showing a model of a circle with intersecting chords that illustrates the following…

Two students drawing geometrical shapes on the chalkboard. A student is also using a scale while students sitting at desks take notes.

Geometry

Two students drawing geometrical shapes on the chalkboard. A student is also using a scale while students…

The circle of multiples can be used as a practice game in class. The outside ring are multiples and the inner ring are factors. n represents the other factor. The teacher points to a set of numbers and students replace n with the other factor. For instance, if 72 is chosen, 8 is the factor, and n would be 9.

Multiples

The circle of multiples can be used as a practice game in class. The outside ring are multiples and…

An illustration showing a model of a circle with an exterior angle formed between a tangent and a secant that illustrates the following geometric relationship: a:t = t:b, t&sup2 = ab

Model Of Geometric Relationships In A Circle

An illustration showing a model of a circle with an exterior angle formed between a tangent and a secant…

An illustration showing a model of a circle with angles formed between tangents and secants that illustrates the following geometric relationship: t&sup2 = (a + b)(a - b).

Model Of Geometric Relationships In A Circle

An illustration showing a model of a circle with angles formed between tangents and secants that illustrates…

An illustration used to "square a circumference".

Square And Circle

An illustration used to "square a circumference".

An illustration used to "square a cirleplane".

Square And Circle

An illustration used to "square a cirleplane".

Illustration of a sector of a circle. A sector is the space between an arc and two radii drawn to the extremities of the arc.

Sector Of Circle

Illustration of a sector of a circle. A sector is the space between an arc and two radii drawn to the…

Illustration of a sector of a hollow circle. A sector is the space between an arc and two radii drawn to the extremities of the arc.

Sector Of A Hollow Circle

Illustration of a sector of a hollow circle. A sector is the space between an arc and two radii drawn…

Labeled shapes showing various measures of form.

Measures of Form

Labeled shapes showing various measures of form.

Four different types of forms or shapes: right triangle, isosceles triangle, rectangle, and circle.

Shapes

Four different types of forms or shapes: right triangle, isosceles triangle, rectangle, and circle.

Illustration showing that a circle may be considered as made up of triangles whose bases form the circumference.

Triangles Making Up A Circle

Illustration showing that a circle may be considered as made up of triangles whose bases form the circumference.

Three students playing a math game in the classroom, throwing beanbags into a circle and doing multiplication problems on the chalkboard.

Math Game

Three students playing a math game in the classroom, throwing beanbags into a circle and doing multiplication…

"Shows how the head advances over half the circumference of the circle which it has to describe, and secures the end of the thread against the support."

Caterpillars of the Cabbage-Butterfly

"Shows how the head advances over half the circumference of the circle which it has to describe, and…

"Use the entire blossom mingled with buds and green leaves, all short stemmed, not longer than three or four inches. Bind the stems with string on a circle made of a piece of willow or some other pliable material, and be sure to removed the thorns from all them stems before weaving the wreath." -Beard, 1906

Wreath of roses

"Use the entire blossom mingled with buds and green leaves, all short stemmed, not longer than three…

"The <em>Crinoidea</em>, which belong to the family of starfishes, are mostly attached to marine rocks by a sort of root, having a long, flexible stem, which enables them to execute movements in the circle limited only by the length of this stem."

Pentacrinus Caput Medusae (Muller)

"The Crinoidea, which belong to the family of starfishes, are mostly attached to marine rocks…

"The whole internal cavity in these animals is occupied by little, white tubes. The mouth opens at the extremity of the body; it forms a sort of funnel, surrounded like a crown with an elegant circle of tentacula."

Cristatella Mucedo (Cuvier)

"The whole internal cavity in these animals is occupied by little, white tubes. The mouth opens at the…

Illustration showing a circle formed by the intersection of a plane perpendicular to the axis of the cone.

Circle

Illustration showing a circle formed by the intersection of a plane perpendicular to the axis of the…