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If the faces be all equal equilateral triangles the solid is termed the "regular" tetrahedron.

139Arrangements connected with Claus' formula are obtained by placing six tetrahedra on the six triangles formed by the diagonals of a plane hexagon.

85If there were such a thing as a triangle contained by absolutely straight lines, its three angles would no doubt measure what Euclid says; but straight lines and true triangles nowhere exist in reruns natura.

76The investigation of triangles and other figures drawn upon the surface of a sphere is all-important in the sciences of astronomy, geodesy and geography.

42The spherical triangles ABC, ABC

44The quadrilateral, for instance, consists of two triangles, and its area is the product of half the length of one diagonal by the sum of the perpendiculars drawn to this diagonal from the other two angular points.

31The arc measured was 3° 7' 3" in length; and the work consisted of two measured bases connected by a series of triangles, one north and the other south of the equator, on the meridian of Quito.

32contains problems of finding rational right-angled triangles such that different functions of their parts (the sides and the area) are squares.

32The artificial character of the diction renders it in emotional passages stilted and even absurd, and makes Canning's clever caricature - The Loves of the Triangles - often remarkably like the poem it satirizes: in some passages, however, it is not without a stately appropriateness.

32Wittich in 1584 made known at Cassel the calculation of one case by this prosthaphaeresis; and Justus Byrgius proved it in such a manner that from his proof the extension to the solution of all triangles could be deduced.3 Clavius generalized the method in his treatise De astrolabio (1593), lib.

32Now, madam, these triangles are equal; please note that the angle ABC...

32If there are no redundant members in the frame there will be only two members abutting at the point of support, for these two members will be sufficient to balance the reaction, whatever its direction may be; we can therefore draw two triangles, each having as one side the reaction YX, and having the two other sides parallel to these two members; each of these triangles will represent a polygon of forces in equilibrium at the point of support.

20In "geodesy," and the cognate subject "figure of the earth," the matter of greatest moment with regard to the sphere is the determination of the area of triangles drawn on the surface of a sphere - the so-called "spherical triangles"; this is a branch of trigonometry, and is studied under the name of spherical trigonometry.

21Warren type in which the bracing bars form equilateral triangles, the Whipple Murphy in which the struts are vertical and the ties inclined, and the lattice in which both struts and ties are inclined at equal angles, usually 45° with the horizontal.

21The fourth book deals with the circle in its relations to inscribed and circumscribed triangles, quadrilaterals and regular polygons.

21The spherical A isosceles triangles AJB, BJC are con gruent, and we see that AB can be brought into the position BC by a rotation about the axis OJ through an FIG.

21If we construct a, the spherical triangles ABC,

21In a similar manner, four covertical equilateral triangles stand on a square base.

21Five equilateral triangles covertically placed would stand on a pentagonal base, and it was found that, by forming several sets of such pyramids, a solid could be obtained which had zo triangular faces, which met in pairs to form 30 edges, and in fives to form 12 vertices.

21That the triangle could give rise to no other solid followed from the fact that six covertically placed triangles formed a plane.

21Cayley gave the formula E + 2D = eV + e'F, where e, E, V, F are the same as before, D is the same as Poinsot's k with the distinction that the area of a stellated face is reckoned as the sum of the triangles having their vertices at the centre of the face and standing on the sides, and e' is the ratio: " the angles subtended at the centre of a face by its sides /2rr."

21The snub cube is a 38-faced solid having at each corner 4 triangles and I square; 6 faces belong to a cube, 8 to the coaxial octahedron, and the remaining 24 to no regular solid.

21The fact that the interior angles of all triangles are equal to two right angles is not part of the definition, but is universally true.

22This triangle, for the convenience of calculation, we divide into two right-angled triangles.

10(i) The figure may be divided into triangles.

10Of these two triangles, shown in fig.

10Legendre was also the author of a memoir upon triangles drawn upon a spheroid.

10The " new invention in Denmark " to which Anthony Wood refers as having given the hint to Napier was probably the method of calculation called prosthaphaeresis (often written in Greek letters irpooOa4aipeats), which had its origin in the solution of spherical triangles.

102 The method consists in the use of the formula sin a sin b=2 {cos(a-b)-cos(a+b)l, by means of which the multiplication of two sines is reduced to the addition or subtraction of two tabular results taken from a table of sines; and, as such products occur in the solution of spherical triangles, the method affords the solution of spherical triangles in certain cases by addition and subtraction only.

10formulae, not involving the conception of an angle as generated by rotation, belong to this stage; the additional geometrical idea involved being that of the proportionality of the sides of similar triangles.

00The regular octahedron has for its faces equilateral triangles; it is the reciprocal of the cube.

00The city park system includes Ottawa Park (280 acres), Bay View Park (202 acres), Riverside Park (118 acres), Central Grove Park (loo acres), Collins Park (90 acres), Walbridge Park (67 acres), with a zoological collection, Navarre Park (53 acres), several smaller parks and triangles, and a boulevard, 18 m.

00The theorem, then, seems to have been arrived at by induction, and may have been suggested by the contemplation of floors or walls covered with tiles of the form of equilateral triangles, or squares, or hexagons.

00[3] Thales discovered the theorem that the sides of equiangular triangles are proportional.

00In applying tests of memory, it may be legitimate to allow a candidate to pass who answers correctly from 30 to 50% of the questions; such an allowance if applied to a test of capacity, such as the performance of a sum in addition, the solution of triangles by means of trigonometrical tables, or the translation of an easy passage from a foreign language, appears to be irrational.

00Even these had been preceded by earlier Pythagorean constructions envisaging the divine life in divine triangles.

00This follows from the fact that all such triangles are necessarily similar.

00The relation between the three forces acting on any particle, viz, the extraneous force and the tensions in the two adjacent portions of the string can be exhibited by means of a triangle of forces; and if the successive triangles be drawn to the same scale they can be fitted together so as to constitute a single force-diagram, as shown in fig.

0016) represent the two forces, AD their resultant; we have to prove that the sum of the triangles OAB, OAC is D equal to the triangle OAD,

00the triangles OAB, RHK are similar, and if the perpendiculars OM, RN be drawn we have HK .OM~rAB.

00successive triangles in the diagram of forces may be constructed in the order XYZ, ZXA, AZB.

00The composition of finite rotations about parallel axes is, a particular case of the preceding; the radius of the sphere is now infinite, and the triangles are plane.

00Further, at any one of the centres of load let PL represent the magnitude and direction of the gross load, and Pa, Pb the two resistances by which the piece to which that load is applied is supported; then wifl those three lines be respectively the diagonal and sides of a parallelogram; or, what is the same thing, they will be equal to the three sides of a triangleS and they must be in the same plane, although the sides of the polygon of resistances may be in different planes.

00Or because of the proportionality of the sides of triangles to the sines of the opposite angles, sin TOB: sin TOA: sin AOB:: a: ~: y, (8 A~

00For, let PiP~ cut the line of centres at I; then, by similar triangles, CiP2 C1Pi IC2: ICi; (24)

00that the triangles CbD and DcE are .~

00From the great Indus series of triangles bases have been selected at intervals which have supported minor chains of triangulation reaching into the heart of the country.

00identity and difference among ideas, as when we say that " blue is not yellow "; or (b) with mathematical relations, as that " two triangles upon equal bases between two parallels must be equal "; or (c) in assertions that one quality does or does not coexist with another in the same substance, as that " iron is susceptible of magnetical impressions, or that ice is not hot "; or (d) with ontological reality, independent of our perceptions, as that " God exists " or " I exist " or " the universe exists."

00The names of these five solids are: (r) the tetrahedron, enclosed by four equilateral triangles; (2) the cube or hexahedron, enclosed by 6 squares; (3) the octahedron, enclosed by 8 equilateral triangles; (4) the dodecahedron, enclosed by 12 pentagons; (5) the icosahedron, enclosed by 20 equilateral triangles.

00If three equilateral triangles be placed at a common vertex with their covertical sides coincident in pairs, it is seen that the base is an equal equilateral triangle; hence four equal equilateral triangles enclose a space.

00Svo - Kat- rpoieKovra, thirty-two), is a 32-faced solid, formed by truncating the vertices of an icosahedron so that the original faces become triangles.

00The pentagons belong to a dodecahedron, and 20 triangles to an icosahedron; the remaining 60 triangles belong to no regular solid.

00alternate remaining bread into triangles and fill the ramekins with these, alternating with layers of chocolate and orange rind.

00congruent triangles in formal geometric proofs.

00neighboring pixels is divided into two triangles.

00Put the two triangles together to make a parallelogram.

00Second order pentagons are made from six first order pentagons plus five first order triangles.

00So far, we have found three regular polyhedrons made with triangles.

00renters insurance rate triangles warn tags writing pads.

00Outline shapes Outline shapes, such as triangles, rectangles and parallelograms, consist of a series of straight-line segments between the vertices.

00They use sine, cosine and tangent in right-angled triangles when solving problems in two dimensions.

00The relative direction of beta strands is shown by the orientation of the triangles.

00supramolecular architectures such as double and triple helices, triangles and grids can be prepared.

00The first border in finer tesserae is of blue triangles set in light gray.

00The rear view one is on the straight, again in Kennington, toward the flood with the warning triangles in the picture.

00The main lithic finds include microliths, particularly scalene triangles, and irregularly shaped picks made mostly of cherty limestone.

00Whenever triangles turn up, you need to know trigonometry to deal with them.

00To achieve smooth results, any triangles which share a common vertex should have the same normal vector at that vertex.

00vertices of the triangles form irregularly spaced nodes.

00"Napier's rules of circular parts," which include the complete system of formulae for the solution of right-angled triangles, may be enunciated as follows.

00The arc measured was 3Ã‚° 7' 3" in length; and the work consisted of two measured bases connected by a series of triangles, one north and the other south of the equator, on the meridian of Quito.

00Warren type in which the bracing bars form equilateral triangles, the Whipple Murphy in which the struts are vertical and the ties inclined, and the lattice in which both struts and ties are inclined at equal angles, usually 45Ã‚° with the horizontal.

00In geodetic measurements the dimensions of the triangles vary with the temperature of the earth, but these variations in the same region of the earth are smaller than the variations of the temperature of the air, less than 10Ã‚° C. Adopting as a co-efficient of dilatation of the earth's crust 0.000002, the variations of the distances are smaller than the errors of measurement (see Geodesy).

00P the octahedron faces; the latter are particularly well defined and have the form =% of equilateral triangles (fig.

00The snub dodecahedron is a 92-faced solid having 4 triangles and a pentagon at each corner.

00The stars are the apexes of what wonderful triangles!

00Time you train rule renters insurance rate triangles warn tags writing pads.

00These patches represent the triangles of the sparse polyhedral approximation to the original shape surfaces.

00A variety of supramolecular architectures such as double and triple helices, triangles and grids can be prepared.

00The vertices of the triangles form irregularly spaced nodes.

00Three basic waves, sine, triangles and square, generated by a locked waveform generator, are applied to shape the display.

00Some parents like to make a row of the days of the week and out beside each day draw several squares, triangles, or circles for each time their child uses the potty.

00For proper presentation, remove the crust from the sandwiches and cut them neatly into small squares or triangles.

00Identifying shapes: Knowing how to identify circles, squares, triangles, and other basic shapes is an important part of getting ready for school.

00At the top of the tri-fold, unfold two triangles (these will come around the sides of your baby's waist).

00Work triangles have three main components.

00Work triangles should also have no more than nine feet and no less than four feet between stations.

00Keep in mind that it is not unusual to have two or more triangles in a large kitchen.Now visualize how the space will be used.

00If it is expected that more than one person will be working in the kitchen, overlapping work triangles should be avoided.

00For a clown, add bright hued triangles around the eyes and color your nose red.

00For a mime, add black triangles around the eyes or paint small teardrops on the corner of the cheekbones.

00Cut them into circles, diamonds or triangles, or cut the edges with a fancy pinking shear that mimics the shape of the border on your page.

00Shapes: Some patterns feature shapes such as circles, squares and triangles.

00Create a banner for your layout by cutting small triangles from patterned paper.

00Take a red piece of cardstock and accent it with black triangles or circles that focus the attention on the picture.

00Heart shapes, stars, hexagons, ovals, rectangles, and triangles are all more unique options for a stunning cake.

00You can also make a 2D deer head cake by baking a small round cake and adding two triangles for the ears.

00Some choose to mount the diamonds in a circle, while others choose triangles or other geometric shaped pendants to display the sparkling diamonds.

00Colorful shapes: Some necklaces contain various shapes such as circles, squares or triangles.

00If you are unable to free your vehicle, put out emergency flares or reflective triangles to alert other drivers that you are stuck.

00One way to test graphics power is to see how many triangles can be processed in a second.

00Using PowerVR SGX540 graphics, this phone can handle up to 90 million triangles per second.

00Contrast this to the 28 million triangles from the iPhone 4, 22 million triangles on the Google Nexus One, and the 14 million triangles on the Palm Pre Plus.

00The triangles in a family system usually interlock in a way that maintains family homeostasis.

00Common family triangles include a child and his or her parents; two children and one parent; a parent, a child, and a grandparent; three siblings; or, husband, wife, and an in-law.

00Family therapists typically look for the level and types of emotions expressed, patterns of dominance and submission, the roles played by family members, communication styles, and the locations of emotional triangles.

00Once you understand how to create a Golden Rectangle, you can see how easy it is to also create Golden Triangles within a square or rectangle.

00You should be able to take any Golden Rectangle or Golden Triangle building and divide the building into infinite rectangles and triangles.

00The Great Pyramid at Giza is composed of Golden triangles.

00Education.com has sheets that help children identify different types of triangles as well as a variety of reviews in a fun game-like format.

00Unlike traditional origami, where a single piece of paper is used to create objects, with 3D origami projects, multiple paper triangles become the building blocks of each piece.

00Continue folding a second time on each end of the points until the first piece of paper has two slightly offset triangles pointing in opposite directions.

00Lay the first set of triangles on top of the second set at a 90 degree angle so that it resembles a pinwheel.

00This will help you make uniform squares, triangles, and strips with ease.

00Curran's remarkable dedication to the two-piece has led to a wide variety of halters, bra tops, ring tops, and triangles in sensible neutral colors.

00This look also features rhinestone details, this time in the shape of triangles.

00No Strings: One of the more daring styles, this type of bikini features a 'top' that is made of tiny triangles of fabric over the nipples.

00The top is a triangle style with adjustable sliding triangles that tie at the neck and back.

00Nikki Gauze Bikini has a triangle top that has adjustable sliding triangles and ties at the neck and back.

00For comparison, take the Brazilian Bikini, and make it less conservative by removing another inch or two from those tiny fabric triangles that barely cover the bosom.

00The top is fixed triangles with a sexy ring in the middle and removable cups.

00Charcoal on one side and lime on the other, this top has built-in cups and fixed triangles.

00Whoever thought that four little triangles of fabric could cause such a big stir?

00String bikinis are made from triangles of fabric which are loosely connected by string ties.

00One piece thong swimsuits can be just as daring as their bikini counterparts, which can sport tops in many styles including halters, bandeaus, tanks and triangles.

00The tops, ranging from A-D cup or small to extra-large, consist of triangles and come in several styles and colors.

00Wicked Weasel Bikinis only offers top pieces in one style, triangles, although you do have a wide range of sizes to choose from so you can play around with the coverage.

00You can also choose the actual fabric width of your triangles.

00Although a basic bikini is little more than a collection of triangles, you should still work with a pattern when sewing your own suit and you should be prepared to make several adjustments along the way.

00Either the strings or the triangles will be prone to movement against the force.

00Since triangles are bottom heavy, the goal is to create balance on top.

00Shapes - Younger children will want to start with squares, circles, triangles, rectangles and ovals but older children could use more advanced geometric shapes to reinforce their geometry lessons.

00They are woven in patterns, such as geometric triangles or stripes.

00Cut the fabric into large triangles and sew one end down the back of a long-sleeved shirt.

00Some people prefer a combination of round stones and baguettes (rectangles), round stones only, or trilliums (triangles).

00Cut in halves, squares, or triangles and serve as a snack or with breakfast.

00Start with basic shapes, such as triangles, squares, rectangles, and circles.

00The ancient tangram, a Chinese puzzle consisting of five triangles, a square, and a rhomboid, is still popular around the world today.

00Each letter is composed of the same lines as circles, triangles, and squares.

00From love triangles to the threat of death, the stories within these movies capture some of the most poignant moments of true love in film history.

00Sidi acknowledges the fact that men and women have different shaped feet: rectangles and triangles, to be exact.

00The intricacies of soap opera love triangles on Passions are always presented in an over the top, outlandish manner that works well within this medium.

00Power struggles at Newman Enterprises are as enticing to fans as the love triangles, illegitimate children and affairs that also populate the landscape.

00Dies are available that can produce squares, rectangles, triangles and several other geometric shapes.

00Add triangles of felt, buttons, rounded pieces of felt for scales, and other embellishments to personalize your dinosaur.

00In their simplest, sexiest iteration, they consist of two tiny fabric triangles joined by strings, ribbons or fabric sides and designed for minimal coverage.

00When the photo is uploaded, you will see it blown up large under an elf hat with blue dots and gray triangles along the sides, outlining the shape of a head.

00The croutons are one-inch sliced triangles lightly fried in oil and vinegar.

00For figures of more than four sides this method is not usually convenient, except for such special cases as that of a regular polygon, which can be divided into triangles C by radii drawn from its centre.

01Take any point P in the latter, and form triangles by joining P to each of the sides AB, BC, ...

01Then (as the difference of two triangles) area ABCD - (h cot 2 (h cot -a)2 2(cot ¢- cot 4) 2(cot +cot 0) (ii) If 0=0, this becomes tan area (h + a tan 0) 2 - a tan 0.

01Then AD =EB = 2h, and the triangles AVD and BVE are equal.

01Over and above these there are other marks, crosses, triangles, &c., of which more than a hundred have been described and figured by different authors, each with its interpretation; and in addition the back of the hand has its ridges.

01

The word usage examples above have been gathered from various sources to reflect current and historial usage. They do not represent the opinions of YourDictionary.com.