Basic Geometry Equations and Examples

Basic Geometry Equations and Examples Mastering Basic Equations of Geometry ChaptersThe Basic ShapesCalculating TrianglesCalculating QuadrilateralsCalculating PolygonsCalculating CirclesSome people might say that geometry is in no way a ‘sexy’ subject; really, as a general rule, calculating angles, volumes and areas is seldom considered enticing or fun.Could the opposite be true?Over the last 10 years, we’ve seen mathematics creeping into films and television shows; The Big Bang Theory is a prime example of such. Granted, equations are not central to the plot and, quite frankly, only the first few shows were math-heavy. After that, algebraic work popped up only occasionally.Still, it is nice to see complex calculations playing out in a popular arena, and it’s even better that both male and female characters take part in tweaking the equations; a  mere 20 years ago, cinematic mathematicians could only be male!Now it’s your turn to master basic geometry equations and you want the most efficient way of doing so. Or maybe you’re a fan of Descartes an d wish to take Cartesian geometry to the next level but you need a solid foundation, first.Your Superprof wants to help you get a good grasp of fundamental geometrical formulas; grab your squares and compasses… we’re off! MyriamMaths Teacher 5.00 (13) £20/h1st lesson free!Discover all our tutors MarkMaths Teacher 5.00 (5) £200/h1st lesson free!Discover all our tutors Dr parikhMaths Teacher 5.00 (8) £40/h1st lesson free!Discover all our tutors KamalMaths Teacher 5.00 (9) £30/h1st lesson free!Discover all our tutors PetarMaths Teacher 5.00 (8) £40/h1st lesson free!Discover all our tutors GowsikaMaths Teacher 5.00 (5) £15/h1st lesson free!Discover all our tutors RubenMaths Teacher 5.00 (1) £15/h1st lesson free!Discover all our tutors ConorMaths Teacher 4.75 (4) £30/h1st lesson free!Discover all our tutorsThe Basic Shapes How many geometric figures can you find in this pattern? Image by monicore from PixabayYou might be tempted to think ‘circle’, ‘triangle’ or ‘square’ and you’d be absolutely correct.Each of those geometric shapes fall into one of these four general categories:Triangles have three sides; the sides may be of equal length (equilateral triangle) or all different length (scalene triangle).A quadrilateral is any four-sided polygon. Those would be rectangles, squares, rhombuses, diamonds…the parallelogram, a shape that has 2 pairs of equal sides, is also a quadrilateralPolygons: literally ‘many sides’. These shapes can be triangles, hexagons, pentagons… all of those ‘gons’ are polygons. Essentially, anything that has straight sides is called a polygon.Circles are a class onto themselves because they have no straight linesTheir unique characteristics include:Squares have four equal sides and four right anglesRectangles have two pairs of equal sidesA trapezoid has on ly one pair of parallel sidesA trapezium has no sides of equal lengthRhomboids: opposite sides and opposing angles are equalThe isosceles triangle has two equal sidesRight triangles have one 90-degree angle opposite of the hypotenuseEach of these shapes has its own formula to calculate its perimeter, area and angles. Some you may be familiar with, such as the Pythagorean theorem while others are perhaps a bit less memorable.Let’s take a look at them now.Do you need help with your geometry studies? Perhaps you could find a geometry tutor…Calculating TrianglesStarting with the shapes of the fewest sides (but sometimes the most complicated formulas), we tackle geometric formulas head-on!The simplest formula for the perimeter of any triangle is a+b+c, with each letter representing a side. It is beautiful in its simplicity and easy to work with, provided you know each side's length.Let’s say your triangle has these measurements: a = 3 inches, b = 4 inches and c = 5 inchesIts perime ter would then be 3+4+5=12 inches.Clearly, this is a triangle is neither equilateral nor isosceles; nor is it a right triangle. How would we calculate the perimeter if only two values, the bottom and one side, are given?In such a case, we have to draw on Pythagoras’ theorem: a2+b2=c2. You remember that one, right?First, draw a line from the triangle’s peak straight down to its base. This line, h, should be perpendicular to the base, thereby forming two 90-degree angles â€" one on each side of the line.You now have two right triangles, one of which has a measurement for both a and b. From there, it is a simple matter to plug known values into the theorem (don’t forget to square them!) and find your missing value.Let’s try it with a fictitious triangle:a = unknown b = 5 c = 7a2 * 52 = 72a2 * 25 = 49 the unknown value must stand alone on one side of the equationa2 = 49 â€" 25 move 25 to the other side of the equal sign, subtracting it from the given value of ca2 = 24Now you hav e to calculate the square root of 24 to find the value of 'a', which is 4.898. Once you've calculated the perimeter of one right triangle, you must calculate the second to get the dimensions of the original triangle.Congratulations! You now know how to calculate the perimeter of any triangle! This and similar triangles signs are used to urge caution on roadways Image by Gerd Altmann from PixabayCalculating Triangles’ AreaWhile perimeter calculation is a rather simple endeavour, figuring the area of a triangle is a bit more involved.If values are given for all three sides, you may apply Heron’s Formula:area = square root of [s(s-a)(s-b)(s-c)], with 's' being the semi-perimeter, that is (a+b+c)/2It only looks complicated; remember that, when working with a formula, you only need to plug in known values to solve for the unknown. When thought of in that way, the Hero’s Formula, as it is also called, is pretty easy!Now, for ‘area of triangles’ equations where one or more values are unknown.If you know only the value of the triangle’s base and its height, you may apply: area = ( ½) * b * hIf only the length of two sides and the degree of the angle joining them are known, you would use trigonometry to find the missing values. The basic formula is:Area = ( ½) * a * b * sin C Keep in mind that lowercase letters signify line measurements while uppercase letters represent angles.If you only know the values of sides a and c, you would plug them in and calculate sin B. Likewise, if you know b and c, you would employ sin A to get your triangle’s area.Why not practise those for a while before moving on... A=a2 and for rectangles, it is A=l * w. Simple, right?Things start getting complicated when we get into parallelograms and trapezoids; to solve both of those equations, you will need to know the height of the shape (h) an d the length of the base (b) â€" the line at the bottom.Once you know those values, choose the appropriate formula for the shape:b * h = area of parallelograms ( ½)(a+b) * h = area of trapezoids, where  â€˜a’ represents the side opposite of ‘b’.Quadrilaterals may just be the easiest shapes to work with. If you need extra practice, there are plenty of resources online where you can find geometry worksheets and equations to sol ve.Calculating PolygonsWhether you are confronted with an apeirogon (a polygon with an infinite number of sides) or the more familiar hexagon, you need to know how to calculate its perimeter and area.Luckily, apeirogons are only hypothetical; imagine having such a figure to calculate an area for!If your polygon’s sides are all the same length, you can apply P=n * v, where ‘n’ is the number of sides and ‘v’ is the value of each side.If said polygon’s side are not all the same length, you will have to add up those values to get its perimeter. The stop sign is perhaps the most renown regular polygon Image by Walter Knerr from PixabayCalculating Areas of PolygonsThere are several ways to realise the value of any polygon’s area, some of which involve calculations for triangles.First, we tackle the equations for a regular polygon; one whose sides are all the same length. Before we can start any ciphering, we have to determine the polygon’s radius.That involves drawing a circle inside the polygon in such a manner that the circle’s perimeter touches the polygon’s perimeter. This is called an inscribed circle. Once we know that radius’ value, we can apply this formula:A = ½ * p * rFormulae get more complicated the more sides the polygon has.Let’s say the number of sides is represented by ‘n’ and sides by ‘s’. The radius, also called apothem, is designated ‘a’. Of course, ‘A’ represents ‘area’, yielding a formula that looks so:A = ns/4 v 4-s2From here, the formulas get ever more complex. Do they l eave  you struggling with the basics of geometry? You can refer to our complete guide!Calculating CirclesCircles involve neither angles nor lines and their perimeters are called ‘circumference’. However, their calculations do require at least a line segment which is instrumental to any formula for circles.Oddly enough, it seems that the formula for calculating areas of circles is more renown than perhaps for any other geometric shape: pr2, or pi * r2Surely you know/remember that pi (p) has a value of 3.1415...The less-renown formula concerning circles, the one for calculating circumferences is: 2 * p * rBear in mind that these are formulae for calculating the area and perimeter of two-dimensional shapes; once they gain an additional dimension â€" they become 3-D shapes and merit a calculation of volume as well as area and perimeter.Let’s not go off on a tangent, here; we’re quite happy to provide formulas for these basic geometric constructions...But you don’t have to stop here; latch on to our beginner’s guide to geometry!

The Life of Georgia OKeeffe

The Life of Georgia O’Keeffe Georgia O’Keeffe, the Mother of American Modernism ChaptersHumble BeginningsEntering the Art World in EarnestGeorgia O’Keeffe, a Mother of Art MovementsWhen one thinks of American painters, names like Warhol and Pollock readily come to mind. Perhaps, if one is a fan of landscape painting, the commercial success story, Thomas Kinkade, might spark.Each of these names and others are attached to a distinctive painting style that shaped the art culture in 20th Century America and, to an extent, western art as a whole.One particularly influential painter is usually dismissed; often, she is not even considered at all: Georgia O’Keeffe.This American painter with a distinctly Irish surname, so dramatically photogenic that she was a work of art in herself, quietly occupied desert spaces and painted what she saw.Her mastery of colour and nuance; the sensuality depicted in her portrayals of flowers in close-up was at the centre of a polarization of artistic opinion.Disdaining figurative elements altogether, she repeatedly averred that her wo rk was purely representational â€" no matter how often the lushness of her lilies was said to be symbolic of genitalia.Frail in body but strong in spirit, cultured and socially connected but choosing the life of a bohemian; through her art, Georgia was proclaimed a mother while being a mother to none.Today, Superprof examines the mass of contradictions that was Georgia O’Keeffe. FernandoPainting Teacher 5.00 (8) £30/h1st lesson free!Discover all our tutors EmmaPainting Teacher 5.00 (5) £75/h1st lesson free!Discover all our tutors DavidPainting Teacher 4.75 (4) £40/h1st lesson free!Discover all our tutors NatashaPainting Teacher 5.00 (7) £37/h1st lesson free!Discover all our tutors StefaniaPainting Teacher 5.00 (4) £40/h1st lesson free!Discover all our tutors SakinaPainting Teacher 5.00 (1) £12/h1st lesson free!Discover all our tutors LorrenPainting Teacher 5.00 (2) £18/h1st lesson free!Discover all our tutors Off2themoonholliePainting Teacher £8/h1st lesson free!Dis cover all our tutorsHumble Beginnings This deceptively simple representation of what Georgia called My Shanty is an example of Precisionism By Georgia O'Keeffe via WikipediaGeorgia Totto O’Keeffe, the second of seven children in her family, was born in 1887, in a Wisconsin farmhouse. She was named after her maternal grandfather, a Hungarian count named George Totto.Life on the dairy farm was busy but special emphasis was placed on the children’s education. When Georgia was of school age, she was bundled off to the Town Hall School in Sun Prairie to learn the Three Rs: reading, writing and arithmetic.Additionally, Mom saw to it that her girls were educated in the arts; she sent daughters Ida, Anita and Georgia to Sara Mann, a watercolour painter of some renown in their small township.By age 10, little Georgia knew she wanted to be an artist.But first, she had to complete her basic education. She boarded at Sacred Heart Academy in Madison and, when the family relocated to Virginia so Dad could pursue a business interes t, Georgia finished her secondary education at Chatham Episcopalian where, once again, she was a live-in student.Casting about for the best art study programmes, Georgia settled on The School of the Art Institute in Chicago, where she consistently placed at the top of her class.After a bout of typhoid fever that set her studies back a whole year, she moved to New York City in search of a broader art curriculum. She found it at The Art Student’s League, a school founded with the artist in mind: no set curriculum, flexible classes and, most importantly, reasonably priced.Her oil on canvas, Dead Rabbit with Copper Pot, won her a scholarship to their summer school in Lake George, where students could practise ‘plein air’ painting â€" painting outdoors.Georgia gladly accepted the boost that scholarship provided but, deep inside, she felt a sense of disquiet.That prize-winning still life she had painted smacked of impressionism. Georgia could not help but notice that, overall, her s tudies emphasised mimicry; essentially reproducing likenesses of what was already there. That wasn’t the direction she wanted to go in.She was absolved of travelling further down that path when her parents’ financial and health difficulties translated into her no longer being able to pay for classes. She returned to Chicago and took a job as a commercial artist.That might have been the end of Georgia’s art education except for the fact that, once again stricken with a disease â€" measles, this time, she had to abandon her post. She returned to the family fold to convalesce.For four years, Georgia did not paint at all; the smell of turpentine made her sick.Georgia O’Keeffe is one female artist among many; discover a world full of remarkable women painters... How could Georgia paint when the very smell of her studio sickened her? Image by Bilge Can Gürer from PixabayEntering the Art World in EarnestThe strong smell of mineral spirits might have sickened her but Georgia had no problem with charcoal.Now teaching art at Columbia College in South Carolina, Georgia satisfied her need to create art by sketching charcoal abstractions.Today, art historians all aver that those drawings were all highly innovative but, with no progressive artistic circle to critique her work at the time â€" and apparently not trusting her own judgment, Georgia sought the opinion of her friend and former classmate in New York, Anna Pollitzer.Blown away: to be thoroughly impressed, overwhelmed and excited.Had that idiom existed in 1916, Anna would have thus described her reactions to each depiction Georgia had sent.She repaired posthaste to the 291 Gallery, where photographer and art promoter Alfred Stieglitz saw it as his calling to promote the best, brightest most innovative of artistic expressions.He immediately took possession of Georgia’s sketches, mounting them in a prominent location, where they were sure to be seen. Some weeks later, she strode into his gallery, furious and demanding that he take her charcoals down. He photographed her.A partnership in art was born.At first, because he was married, Alfred was just her patron. She moved back to New York and into the studio he provided her with.Later, as she posed for him, as he photographed her, critiqued and sold her work, they became romantically involved, ultimately marrying, once he had divorced his wife.Through him, Georgia became acquainted with some of the biggest names in American art: Marsden Hartley, Arthur Dove, Paul Strand...Strangely enough, though a contemporary of Mary Cassatt, their paths never crossed.It was Paul’s and Alfred’s photography that triggered in Georgia a desire to explore the most delicate organisms in the most minute detail possible. Her first stab at precisionism, the forerunner of modernism, yielded The Green Apple, formerly titled Green Apple on Black Plate.Other works from that period include:Blue and Green MusicOriental PoppiesRed Canna â€" several paintings, some in watercolours and some in oil paintPetuniaJimson WeedBlack Iris IIIRegrettably, even though she painted many subjects and employed different media, Georgia is renown for her flower paintings and what they might represent. One art historian even averred that one tableau in particular must represent a female in bloom!Georgia adamantly refuted any such correlations, maintaining she was only painting flowers in close-up.Although a contributor to the abstract expressionism art movement, Helen Frankenthaler never had to defend her work from such near-libellous claims. Although lilies were a favourite subject, unlike Van Gogh, Georgia never painted water lillies Image by Couleur from PixabayGeorgia O’Keeffe, a Mother of Art MovementsAlbert Stieglitz worked tirelessly to promote his wife’s art. He featured her canvases in several New York Galleries and a few art museums. He even arranged for a New York Museum to host a retrospective of her work in 1927.By far the greatest boost to her reputation was his dubious claim that an anonymous buyer in France had purchased no fewer than six of her calla lily tableaux. While her work certainly did make an impact in Paris, it is doubtful that a lone art collector spent quite the amount he boasted to have received.Nevertheless, the claim made for good publicity and, from then on, Georgia could command a much higher price for her work.It would seem that his tireless efforts to promote her work must mean that he was wholly devoted to her.Sadly, his extramarital affair plunged Georgia into a deep depression th at left her unable to complete a commission to paint a mural at Radio City Music Hall. She fled the city, finding solace in the wide-open spaces of New Mexico.Unlike Frida Kahlo, who retaliated to husband’s extramarital activities with affairs of her own, Albert’s ongoing affair drove Georgia to a nervous breakdown that left her unable to paint for over a year.Finally pulling herself together after a recuperative sojourn in the Bahamas, she returned to New Mexico, purchased a house in a small town and set to work with renewed vigour.One year after being back to her old self, her husband died. She spent the next three years in New York, settling his affairs after which she permanently relocated to the property she had just purchased. There, she worked continuously until her death, at 98 years old.When her eyes failed her â€" effectively leaving her unable to paint, she reinvented herself as a sculptor through the tutelage of one John Hamilton. Now, rather than seeing her creation s come to life, she could feel them taking shape beneath her hands.Georgia O’Keeffe does not receive near the esteem and publicity as American artists who are male â€" unlike some of them, there are no web pages dedicated exclusively to her art.Still, one can find her canvases, depicting building, flowers and still lifes in New York’s Museum of Modern Art as well as the Metropolitan Museum of Art and, of course, in New Mexico, where she is heralded as a native daughter.Now discover the life and works of Artemisia Gentileschi, the Grande Dame of female artists...