Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Sloping |
SlopingAdjective1. Having an oblique or slanted direction. 2. Having a slanting form or direction; "an area of gently sloping hills"; "a room with a sloping ceiling". Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "sloping" was first used in popular English literature: sometime before 1321. (references) |
(From Wikipedia, the free Encyclopedia)
In mathematics, the slope (or gradient, especially where three or more dimensions are discussed) of a straight line (within a Cartesian coordinate system) is a measure for the "steepness" of said line. This pages focuses on such slopes. With an understanding of algebra and geometry, one can calculate the slope of a straight line; with calculus, one can calculate the slope of a curved line at a point.
The concept of slope, and much of this article, applies directly to gradess or gradients in geography and civil engineering.
Definition of a Slope
It is generally represented by m, and defined as the change in y divided by the corresponding change in x (if the horizontal axis is the x-axis and the vertical axis is the y-axis), often written as:and memorized as "rise over run". (The triangular symbol is the Greek letter delta, commonly used in mathematics to mean "change". So m is equal to the change in y, the vertical coordinate, divided by the change in x, the horizontal coordinate; that is m is the ratio of the changes.) This concept is fundamental to algebra, analytic geometry, trigonometry, and calculus.
Note that it doesn't matter which two points on the line you pick, or in which order you use them: the same line will always have the same slope. Curves have "accelerating" slopes and one can use calculus to determine such slopes.
Example 1
Suppose a line runs through two points: P(13,8) and Q(1,2). By dividing the difference in y-coordinates by the difference in x-coordinates, one can obtain the slope of the line:
The slope is 1/2 = 0.5.
Example 2
If a line runs through the points (4, 15) and (3, 21) then:
Geometry
The larger the slope, the steeper the line. A horizontal line has slope 0, a 45° rising line has a slope of +1, and a 45° falling line has a slope of -1. The slope of a vertical line is not defined (it does not make sense to define it as +∞, because it might just as well be defined as -∞).The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function:
and
(see trigonometry).
Two lines are parallel if and only if their slopes are equal; they are perpendicular (i.e. they form a right angle) if and only if the product of their slopes is -1.
Algebra
If the equation of the line is given in the form
then the slope m can be read off as the coefficient of the x variable. This form of a line's equation is called the slope-intercept form, because b can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis.
If you know the slope m of a line and a point (x0, y0) on the line, then you can find the equation of the line using the point-slope formula:
For example, consider a line running through the points (2, 8) and (3, 20). This line has a slope, m, of (20 - 8) / (3 - 2) = 12. One can then write the line's equation, in point-slope form: y - 8 = 12(x - 2) = 12x - 24; or: y = 12x - 16.
Why Algebra Can't Calculate the Slope of a Curve
In algebra, the slope of a straight line can be calculated with m = Δy / Δx; where one is calculating the slope from the values of the function at different points. However, with a curved line (such as that described by y = x²) the slope changes between each and every point described by said curve and one is thus only able to calculate an average slope, via the above method.For example, the average slope of y = x², from x = 0 to x = 3, is m = 9 / 3 = 3 (which happens to be the actual slope at, and only at, x = 1.5). Should one attempt to use the above formula for a single point, such as (x = 3, y = 9); one then gets m = 0 / 0 ; as the &Delta for both y and x equals zero (see also: division by zero).
Calculus
The concept of a slope is central to differential calculus; which deals with functions whose graph is not a line. Unlike linear functions, the slope of a non-linear function varies at different points. This slope is often referred to as a derivative. To find the slope at a given point on a curve, one must find a line which is tangent to said function, at said point. The slope of said tangential line is equal to the slope of said function at said point.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Slope."
Synonyms: SlopingSynonyms: aslant (adj), aslope (adj), diagonal (adj), slanted (adj), slanting (adj), sloped (adj). (additional references) |
| Domain | Title |
Books |
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Source: compiled by the editor from various references; see credits. | |
| Thumbnail | Description & Credit | Thumbnail | Description & Credit |
 | Sapelo Island National Estuarine Research Reserve. Swash zone at Cabretta(?). Looking south. Georgia's coastline is composed of wide, slowly sloping beaches. Wave action is minimal compared to other areas of the East Coast. Credit: National Estuarine Research Reserve System (NERR). |  | NRCS Conservationists examine a well-established filter strip on this Cass County, Illinois farm. The filter strip captures sediment and excess nutrients leaving the sloping farmland. Credit: Bob Nichols. |
 | Stripcropping on sloping land reduces sheet and rill erosion and resulting sediment. Credit: Glenn Shea. |  | Colonial Village. Houses with sloping terrain at Colonial Village. Credit: Library of Congress. |
 | Ronquieres, cultural and touristical sight : visit the "sloping lock" tower. Credit: Library of Congress. | ||
Source: pictures compiled by the editor from various references; see picture credits. | |||
| Subject | Topic | Quote |
Economic History | Democratic Republic of Congo | The vast, low-lying central area is a basin-shaped plateau sloping toward the west and covered by tropical rainforest. (references) |
Chad | The Ennedi Plateau and the Ouaddaï highlands in the east complete the image of a gradually sloping basin, which descends towards Lake Chad. (references) | |
Source: compiled by the editor from ICON Group International, Inc.; see credits. | ||
| Speaker | Term | Phrase(s) |
Ronald Reagan | 1981-1989 | Beyond those monuments to heroism is the Potomac River, and on the far shore the sloping hills of Arlington National Cemetery with its row on row of simple white markers bearing crosses or Stars of David. |
Source: compiled by the editor from various references. | ||
| "Sloping" is generally used as a lexical verb (-ing form) -- approximately 51.98% of the time. "Sloping" is used about 202 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted) |
| Parts of Speech | Percent | Usage per 100 Million Words | Rank in English |
| Lexical Verb (-ing form) | 51.98% | 105 | 31,781 |
| Adjective (general or positive) | 45.54% | 92 | 34,282 |
| Noun (singular) | 1.98% | 4 | 175,879 |
| Noun (proper) | 0.5% | 1 | 339,140 |
| Total | 100.00% | 202 | N/A |
Source: compiled by the editor from several corpora; see credits.
Expressions using "sloping": sloping roof ♦ sloping surface ♦ sloping trough. Additional references. | |
| Hyphenated Usage | |
Beginning with "sloping": sloping-beamed. | |
Ending with "sloping": steeply-sloping, steep-sloping, upward-sloping. | |
| Source: compiled by the editor from various references; see credits. | |
| The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com. |
| Expression | Frequency per Day |
sloping lot house plan | 14 |
floor sloping | 4 |
home lots plan sloping | 4 |
house lots plan sloping | 3 |
sloping lot home plan | 3 |
erosion in lawn sloping | 2 |
| Source: compiled by the editor from various references; see credits. | |
| Language | Translations for "sloping"; alternative meanings/domain in parentheses. | |
Albanian | me pjerrësi, i pjerrët (arduous, bevel, declivitous, declivous, high pitched, hilly, inclined, leaning, oblique, reclinate, sideling, skew, slant, slanting, slantwise). (various references) | |
Arabic | منحدر (aslope, chute, decline, declivity, descending, descent, dipped, downgrade, downhill, downward, inclined, slanted, slope, sloped, talus), مائل (atilt, awry, bent, bevel, bias, cock eyed, diagonal, inclined, oblique, sidelong, skew, slant, slanted, slantwise, slope, stooping, thwart, titled). (various references) | |
Bulgarian | наклонен (downhill, hanging, inclined, intent, low-grade, oblique, one sided, prone, rampant, ready, scalene, skew, slant, slantindicular, slanting, sloped, splay, supine), полегат (declivous, downhill, inclined, oblique, prone, rampant, slant, slantindicular, slanting). (various references) | |
Chinese | (quite, rather, uneven), 倾斜 (Canting, inclined, inclining, lean, Leaned, leaning, Leant, Oblique, raked, raking, slanted, slanting, slope, sloped). (various references) | |
Czech | svažující se (declivitous, downhill), stoupající (acclivous, ascendant, ascensional, rising, soaring), ležatý, kulatý (rotund, round, rounded), klesající (declinatory, declining, decreasing, descending, failing), šikmý (oblique, skew, slant, slanting). (various references) | |
Danish | det at give hældning (banking, battering), det at gøre skrå (banking, battering). (various references) | |
Dutch | schuin (oblique, obscene, slanting), hellend (inclined), glooiend, aflopend (ending). (various references) | |
Esperanto | dekliva. (various references) | |
Finnish | viettävä (downhill, slanting), kalteva (inclined, leaning, slanting). (various references) | |
French | talutage, penché (slanting), incliné (slanted), en pente (on slope, slanted, slope), couché (slanting). (various references) | |
German | schief (askance, aslant, awry, cockeyed, crooked, distorted, inappropriate, inclinated, leaning, lopsided, not straight, oblique, scalene, skew, slanting, sloppily, squint eyed, tilted, worn-down, wry), abschüssig (abrupt, precipitous, scarped, steep). (various references) | |
Greek | κεκλιμένοσ (acclivitous, alist, inclined), με κλίση, επικλινήσ (atilt, prone, slant), διαμόρφωση πρανών (banking, battering). (various references) | |
Hebrew | משופע (inclined, oblique, slanting), מ"רו י (downhill), תלול (high, lofty, precipitous, sharp, sheer, steep). (various references) | |
Hungarian | lejtõs (askew, aslope, downhill, inclined, steep). (various references) | |
Indonesian | miring (askew, aslant, awry, bias, cant, careen, crazy, oblique, slanting, wry), ereng (leaning), cenderung (disposed, inclimed, lean, prone). (various references) | |
Italian | spiovente (arcing shot), obliquo (askance, askew, cross, devious, indirect, lopsided, oblique, sidelong, sideward, slanting, sloped, slopingly, transversal, underhand, wry), mettere in pendenza (banking, battering), mettere a scarpata (banking, battering), inclinato (a-tilt, broken, downhill, inclined, nodding, oblique, slanting). (various references) | |
Japanese Kanji | 'で肩 (sloping shoulders), 合掌 り (structure with a sloping or rafter roof). (various references) | |
Japanese Katakana | がっしょうづくり (structure with a sloping or rafter roof), なでがた (sloping shoulders). (various references) | |
Korean | 경사짐. (various references) | |
Manx | ughtee (bluff, bluff of coast, sheer, sloping upwards). (various references) | |
Pig Latin | opingslay.(various references) | |
Portuguese | inclinado (apt, aslope, atilt, biased, canted, declivous, downhill, inclinable, incumbent, leaning, minded, oblique, pendulous, prone, ready, slanting, superabundant). (various references) | |
Romanian | povârnit, oblic (aslant, asquint, athwart, bias, diagonal, inclined, oblique, obliquely, sidelong, skew, slanting, splay, squint, transverse), înclinat (aslope, atilt, declivous, inclinable, inclined, lop-sided, minded, oblique, overhung, prone, rising, slanting, splay, squint), în pantã (acclivous, aslope, declivous, downhill, rising, slanting, uphill). (various references) | |
Russian | наклонный (downhill, inclined, oblique, prone, reclinate, sideling, slant, slanting, slantwise, sloped). (various references) | |
Scottish | learg (a sea coast, diver bird, sloping plain), l irig (a moor, moor, sloping hill). (various references) | |
Serbo-Croatian | nagnut (bevel, declivous, inclined, leaning, rakish, sideling, slanting), kos (awry, bevel, lop-sided, oblique, ousel, ouzel, skew, slant, slantindicular, slanting, slantwise, traverse). (various references) | |
Spanish | sesgo (askance, askew, cant, lopsided, slant, slanting, turn, twist, warp, wry), inclinación (bent, bow, careenage, desire, dip, droop, gradient, inclination, incline, inclining, item, leaning, liking, list, mind, nod, nutation, penchant, pitch, propensity, skew, slant, slanting, slope, tilt, tilting, want), en pendiente (aslope, downhill), costanero (coastal, scarped, seaboard), ataluzado (banking, battering). (various references) | |
Swedish | sluttande (cant, declivous, downhill, inclined, prone, slanting). (various references) | |
Turkish | meyilli (aslant, aslope, bias, declivitous, downhill, inclinable, inclined, leaning, oblique, on the slope, prone, skew, slant, slanting, slantwise, subject to, tilting), eğri (aslant, aslope, awry, bent, cambered, cockeyed, crooked, curve, graph, oblique, out of the straight, sinuous, skew, skewed, slanting, slouching, slouchy, tortuous, trajectory, untrue, warped, wry), eğimli (apt, curved, declivitous, inclined, prone, slant, tilting), eğik (bevel, drooping, droopy, inclined, leaning, oblique, on the slope, skewed, slanting, threshold), şevli (bevel, bevelled, splay). (various references) | |
Ukrainian | скошений (bevelled, canting, squint), похилий (askew, aslope, canting, declining, downhill, inclined, oblique, prone, sidelong, slanting). (various references) | |
Vietnamese | nghiêng (askew, asquint, over, sideling, skew, slant, slanting, slantways, slantwise, slopewise), dốc (bias, declivity, gradient, ramp, slope, slopewise, steep, uphill, versant). (various references) | |
Welsh | llethrog (declining, steep), gw+yr (crooked, Gower, oblique). (various references) | |
| Source: compiled by the editor from various translation references. | ||
| Language | Period | Translations |
| Latin | 500 BCE-Modern | clivum, cubans. (various references) |
| Source: compiled by the editor from various references. | ||
| Language | Date | Source | Ezekiel Chapter 41, Verse 26 |
| Greek (transliterated) | 250 BC | Septuagint | Kai qurideV kruptai kai diemetrhsen enqen kai enqen eiV ta orofwmata tou ailam kai ta pleura tou oikou ezugwmena |
| Latin | 405 | Vulgate | Super quae fenestrae obliquae et similitudo palmarum hinc atque inde in umerulis vestibuli secundum latera domus latitudinemque parietum |
| Middle English | 1395 | Wyclif | On whiche the ouerthwert wyndowis, and lickenesse of palmes, on this syde and that syde; in the litil shuldris of the porche, after the sydis of the hous, and breede of wallis. |
| Jacobean English | 1611 | King James | And there were narrow windows and palm trees on the one side and on the other side, on the sides of the porch, and upon the side chambers of the house, and thick planks. |
| Victorian English | 1833 | Webster | And there were narrow windows and palm-trees on the one side and on the other side, on the sides of the porch, and upon the side-chambers of the house, and thick planks. |
| Basic English | 1964 | Ogden | And there were sloping windows and palm-trees on one side and on the other, on the sides of the covered way: and the side-rooms of the house and the ... |
Source: compiled by the editor from various references; see credits. | |||
| Language | Ezekiel Chapter 41, Verse 26 |
| Cebuano | Ug may mga tinakpan nga tamboanan ug mga kahoyng palma sa usa ka kilid ug sa pikas nga kilid, sa mga kiliran sa alagianan: mao kini ang kilirang lawak sa balay, ug ang mga tukmaan. |
| Croatian | Prozori s rešetkama i palme bijahu s jedne i s druge strane na trijemu, u poboènim prostorijama Doma i na nadstrešnici. |
| Danish | Der var gitrede Vinduer og Palmer på Forhallens Sidevægge til begge Sider..." |
| Dutch | En aan de gesloten vensteren waren ook palmbomen van deze en van gene zijde, aan de zijden van het voorhuis; en aan de zijkameren van het huis, en aan de dikke planken. |
| Finnish | Ja eteisen sivuseinissä, kummallakin puolella, oli sisäänpäin avartuvia ikkuna-aukkoja ja palmuja; samoin temppelin sivukammioissa ja porraskatoksissa. |
| French | Il y avait des fenêtres fermées, et il y avait des palmes de part et d`autre, ainsi qu`aux côtés du vestibule, aux chambres latérales de la maison, et aux entablements. |
| German | Und es waren enge Fenster und viel Palmlaubwerk herum an der Halle und an den Wänden. |
| Hungarian | És szoros ablakok és pálmák valának mind egy- s mind másfelõl a tornácz oldalfalain. Így voltak a ház oldalkamarái és a gerendák. |
| Indonesian-Bahasa Sehari-hari | Pada kedua sisi dinding sampingnya terdapat jendela-jendela, dan dindingnya dihias dengan ukir-ukiran pohon palem. |
| Indonesian-Terjemahan Lama | Maka adalah tingkap-tingkap yang sempit dan pokok kurma pada sebelah sini dan pada sebelah sana pada segala iringan rambat dan pada segala petak iringan rumah dan pada papan yang tebal-tebal itu. |
| Italian | Finestre e grate e palme erano da tutt'e due le parti, ai lati del vestibolo, alle celle annesse al tempio e agli architravi. |
| Maori | A he matapihi kuiti, he nikau hoki, i tetahi taha, i tetahi taha, i nga taha hoki o te whakamahau: koia era, ko nga ruma o te taha o te whare, me nga papa matotoru hoki. |
| Norwegian | Og det var vinduer med fast gitter og palmer på forhallens sidevegger, på begge sider, og i husets sidekammere og på tverrbjelkene. |
| Portuguese | Também havia janelas fechadas e palmeiras, de uma e de outra banda, pelos lados do vestíbulo. |
| Rumanian | Erau ferestre kngrqdite, wi deoparte wi de cealaltq erau finici, wi tot awa era wi pe laturile pridvorului, pe odqile lqturalnice wi pe grinzile de sus. |
| Spanish | Había ventanas anchas por dentro y angostas por fuera, y decoraciones de palmeras a uno y otro lado de los costados del vestíbulo, tanto en los cuartos laterales del edificio como en los aleros. |
Source: compiled by the editor from various references; see credits. | |
Misspellings | |
"Sloping" is suggested in spellcheckers for the following: salopian, scoping, sliping, sloging, sloin, spoling. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
| # of Phoneme Matches | Pronunciation | Word(s) rhyming with "sloping" (pronounced slō"ping) |
| 4 | -ō" p i ng | coping, groping, hoping, moping, roping, scoping. |
| 3 | -p i ng | antidumping, aping, backslapping, beeping, bleeping, bookkeeping, bumping, burping, camping, capping, carping, Chipping, chirping, chomping, chopping, clamping, clapping, clipping, clumping, copping, cramping, creeping, crimping, cropping, damping, developing, dipping, draping, dripping, drooping, dropping, dumping, eavesdropping, enveloping, equipping, escaping, flapping, flipping, flopping, galloping, gaping, gasping, gossiping, grasping, griping, gripping, grouping, gulping, handicapping, harping, heaping, helping, hopping, housekeeping, hyping, jumping, keeping, kidnaping, kidnapping, lamping, landscaping, lapping, leaping, limping, lopping, lumping, mapping, mopping, napping, nipping, outstripping, overlapping, overstepping, peacekeeping, peeping, pimping, piping, popping, prepping, propping, pulping, pumping, ramping, raping, rapping, reaping, recapping, recouping, regrouping, reshaping, revamping, ripping, romping, safekeeping, sapping, scalping, scooping, scraping, scrapping, scrimping, seeping, shaping, shipping, shopping, sidestepping, sipping, skimping, skipping, slapping, sleeping, slipping, slumping, snapping, sniping, snooping, sopping, stamping, stepping, stereotyping, stomping, stooping, stopping, strapping, stripping, stumping, swamping, swapping, sweeping, swiping, swooping, tamping, taping, tapping, thumping, tipping, topping, tramping, trapping, tripping, trooping, typing, unwrapping, upping, usurping, videotaping, walloping, Wapping, warping, weeping, whipping, whooping, whopping, wiping, Wiretapping, worshipping, wrapping, zapping, zipping. |
Source: compiled by the editor (additional references); see credits. | ||
Scrabble® Enable2K-Verified Anagrams | |
| Words within the letters "g-i-l-n-o-p-s" | |
-1 letter: gipons, loping, losing, pingos, poling, posing, soling. | |
-2 letters: gipon, glops, lingo, lings, linos, lions, loins, longs, noils, oping, opsin, pingo, pings, pions, polis, pongs, sling, spoil. | |
-3 letters: gins, gips, glop, ions, ling, lino, lins, lion, lips, lisp, logs, loin, long, lops, nils, nips, nogs, noil, oils, pigs, ping, pins, pion, piso, pois, pols. | |
| Words containing the letters "g-i-l-n-o-p-s" | |
+1 letter: longship, pignolis, posingly, slopping, spoiling, spongily, spooling. | |
+2 letters: couplings, longships, pangolins, pignolias, pilotings, pistoling, plainsong, polishing, splodging, sploshing, spoolings, stoppling. | |
+3 letters: collapsing, despoiling, disploding, escaloping, imposingly, nonplusing, oligopsony, opalescing, penologies, penologist, phlogiston, pistolling, plainsongs, polarising, polygonies, polygynies, prolapsing, scalloping, scolloping, splotching, spoliating, sporangial, sportingly, topsoiling. | |
+4 letters: angioplasty, apologising, depolishing, dispeopling, escalloping, lipogeneses, lipogenesis, nonplussing, outsleeping, outspelling, penologists, phenologies, phlogistons, phonologies, phonologist, phylogenies, pigeonholes, plasminogen, polygenesis, postlanding, postulating, promisingly, proselyting, repolishing, shoplifting, slipforming, snowplowing, sporulating, stockpiling, telescoping, vanpoolings. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
| 1. Definition 2. Synonyms 3. Crosswords 4. Usage: Commercial | 5. Images: Photo Album 6. Quotations: Non-fiction 7. Quotations: Speeches 8. Usage Frequency | 9. Expressions 10. Expressions: Internet 11. Translations: Modern 12. Translations: Ancient | 13. Bible Trace 14. Derivations 15. Rhymes 16. Anagrams | 17. Bibliography |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
