Study Guide

Geometry

    Geometry Introduction

  • Introduction to Geometry

    Have you noticed how many grim words start with the letter G?

    Gross, grimy, and gastrointestinal—especially if you just ate a giant helping of goulash (in which case, "gassy" is probably next on the list). And we haven't even mentioned that graveyards, gout, and gerrymandering fall under the G realm as well. Thank goodness Geometry isn't one of them.

    With a little help from Shmoop, Geometry has ditched the grisly G's and joined the good (and way more glamorous) ones. It may have hung out with the wrong crowd in the past, but everyone deserves a second chance. We've helped Geometry clean up its act—and go from gory to glory.

    Now, that doesn't mean geometry is all sunshine and rainbows. Some calculations can be pretty gnarly, and many proofs are downright grueling. Half the time, finding the right answer feels more like a guessing game than actual math.

    Have no fear! That's why Shmoop is here! With glitzy explanations, guffaw-worthy examples, and Gangnam Style dance moves, we'll make the tasteless intricacies of Geometry seem like gingerbread cookies, with 0 Calories and 100% natural Shmoopy goodness.

    Why Should I Care?

    In algebra, you factored expressions and solved equations, but there wasn't really any rhyme or reason as to why. Sure, parts of it were applicable and probably even interesting, but it may also have felt a bit like going through a zero gravity minefield of variables and concepts. Nothing was tied to solid ground.

    If that's the case, then geometry is practically the opposite—it's like a full gravity minefield. On second thought, that might not be the best analogy.

    What we're saying is that geometry is the math course that's most applicable to the real world. (Don't tell the other math courses. They'll only get jealous.) While solving triangles and writing proofs might not be integral parts of day-to-day activities, geometry really is everywhere you look.

    Want to make sure the circular track at school really is a fourth of a mile? Need to find the best angle to prop up a ladder against a wall? With distances, measurements, and angles, geometry's got your back. Areas, volumes, and densities also fall under geometry territory—and that's only scratching the surface.

    At the very least, geometry gives you terms to accurately describe the world around you. Eventually, tables will turn into rectangles, basketballs will become spheres, and instead of tree trunks, you'll see wooden cylinders. Party hats will morph into colorful cones, tires will look like circles, and kites will be…well, kites. Because some things just don't change.

    • What is Geometry?


      The word "geometry" literally means "earth measure." Explorers, cartographers, and topographers have taken care of all that stuff for us, so what's left for geometers?

      A whole lot, actually. Geometry is a way to measure things in the world as opposed to Planet Earth itself. (Can you imagine the size of the ruler you'd need?) It's about using visible figures and mathematical concepts to understand the physical world around us. Sounds thrilling, doesn't it?

      Don't worry. It's not nearly as bad as it seems. Comparing shapes and applying formulas? Piece of cake. Finding side lengths and analyzing angles? Child's play. Proving theorems and performing constructions? Bring it on.

      In fact, compared to the painstakingly exact measurements that cartographers and topographers have to take, geometry is practically a Hawaiian vacation. So come on and join the luau! Make sure to BYOL: bring your own lei.

    • Basic Elements


      The basic elements of geometry are objects you see every day but probably never think about (unlike your Nintendo DS, which you think about every day and still can't seem to find). We're talking about lines, angles, and shapes—and lots of 'em.

      That being the case, geometry will require you to draw sometimes. In fact, one of the best things to do if you're ever unsure about a geometry problem is to draw a picture. You don't have to be the next Claude Monet or Andy Warhol, but we'd advise you to steer clear of the Pablo Picasso or Salvador Dali neck of the woods.

      You'll also use proofs, fact-based arguments that lead to a logical conclusion, to dissect and discover the properties of these shapes. Chances are good that you've probably never written a proof before, so we'll cover exactly what proofs are and how best to tackle them. (Hint: get a running start.)

      While geometry does primarily work in the visible arena, we'll still need the math tools we've been gathering so far. Hopefully addition, subtraction, multiplication, and division go without saying, but we said them anyway just to be safe. Basic algebra will definitely come in handy also—especially using linear equations and manipulating variables.

      We'll also touch on coordinates and the x-y plane (and even the x-y-z plane), so we're hoping you kept the distance formula in a safe somewhere. If not, don't worry your pretty little head, since it comes from the Pythagorean Theorem anyway.

    • Applications


      A lot of tools and concepts in geometry aren't that popular in other areas of math. Sure, graphs are handy and you might see a triangle thrown into a math lesson every once in a while, but when was the last time you saw a cone in an algebra class? No, that leaning tower of ice cream you snuck in on the last day of school doesn't count.

      Geometry is really the awkward third wheel of math courses. If they were siblings, then algebra, trigonometry, and calculus would be a pop rock band sensation and geometry would be the Bonus Jonas. Underrated, yet full of hidden talent and possibility.

      And honestly, geometry has something that none of the other math fields can claim: a solid application to the real world.

      Seriously! How often are you going to find the degree of a polynomial in real life? When have you ever solved a real-world problem by taking the indefinite integral?

      Geometry, on the other hand, has real-life applications all over the place. All around us are lines, shapes, angles, distances, symmetries, congruencies, circles, squares, triangles, volumes, and areas. Textiles and clothing, video game design, and architecture are just a few of the fields that use geometry and apply it to the world (or virtual world) around us.

      And if you've ever helped put together a piece of Ikea furniture, you know how useful geometry (and an Allen wrench) can be.

  • What is Geometry?


    The word "geometry" literally means "earth measure." Explorers, cartographers, and topographers have taken care of all that stuff for us, so what's left for geometers?

    A whole lot, actually. Geometry is a way to measure things in the world as opposed to Planet Earth itself. (Can you imagine the size of the ruler you'd need?) It's about using visible figures and mathematical concepts to understand the physical world around us. Sounds thrilling, doesn't it?

    Don't worry. It's not nearly as bad as it seems. Comparing shapes and applying formulas? Piece of cake. Finding side lengths and analyzing angles? Child's play. Proving theorems and performing constructions? Bring it on.

    In fact, compared to the painstakingly exact measurements that cartographers and topographers have to take, geometry is practically a Hawaiian vacation. So come on and join the luau! Make sure to BYOL: bring your own lei.

  • Basic Elements


    The basic elements of geometry are objects you see every day but probably never think about (unlike your Nintendo DS, which you think about every day and still can't seem to find). We're talking about lines, angles, and shapes—and lots of 'em.

    That being the case, geometry will require you to draw sometimes. In fact, one of the best things to do if you're ever unsure about a geometry problem is to draw a picture. You don't have to be the next Claude Monet or Andy Warhol, but we'd advise you to steer clear of the Pablo Picasso or Salvador Dali neck of the woods.

    You'll also use proofs, fact-based arguments that lead to a logical conclusion, to dissect and discover the properties of these shapes. Chances are good that you've probably never written a proof before, so we'll cover exactly what proofs are and how best to tackle them. (Hint: get a running start.)

    While geometry does primarily work in the visible arena, we'll still need the math tools we've been gathering so far. Hopefully addition, subtraction, multiplication, and division go without saying, but we said them anyway just to be safe. Basic algebra will definitely come in handy also—especially using linear equations and manipulating variables.

    We'll also touch on coordinates and the x-y plane (and even the x-y-z plane), so we're hoping you kept the distance formula in a safe somewhere. If not, don't worry your pretty little head, since it comes from the Pythagorean Theorem anyway.

  • Applications


    A lot of tools and concepts in geometry aren't that popular in other areas of math. Sure, graphs are handy and you might see a triangle thrown into a math lesson every once in a while, but when was the last time you saw a cone in an algebra class? No, that leaning tower of ice cream you snuck in on the last day of school doesn't count.

    Geometry is really the awkward third wheel of math courses. If they were siblings, then algebra, trigonometry, and calculus would be a pop rock band sensation and geometry would be the Bonus Jonas. Underrated, yet full of hidden talent and possibility.

    And honestly, geometry has something that none of the other math fields can claim: a solid application to the real world.

    Seriously! How often are you going to find the degree of a polynomial in real life? When have you ever solved a real-world problem by taking the indefinite integral?

    Geometry, on the other hand, has real-life applications all over the place. All around us are lines, shapes, angles, distances, symmetries, congruencies, circles, squares, triangles, volumes, and areas. Textiles and clothing, video game design, and architecture are just a few of the fields that use geometry and apply it to the world (or virtual world) around us.

    And if you've ever helped put together a piece of Ikea furniture, you know how useful geometry (and an Allen wrench) can be.

  • Major Geometry Themes

    Lots of TV show themes—especially the good ones—get stuck in your head all day long and make you want to do nothing other than sit back, relax, and watch Dorothy, Rose, Blanche, and Sophia plow into their umpteenth cheesecake.

    Well, we believe that geometry deserves its own theme song. We could probably make it happen with some music lessons, but for now we'll have to settle for a slightly less rhythmic method.

    If you want rhythm, try reading these like the Fresh Prince of Bel-Air.

    • Logic


      When you were a wee little child, your parents would take you to the neighborhood park to play. You'd swing and slide and climb on the jungle gym. Once in a blue moon, you brought colored chalk with you and played tic-tac-toe with your friends on the sidewalk. That was the closest you ever got to merging logic and fun.

      We aren't going to claim that the logic in geometry is as fun as a playground. Very few things in life beat the sheer joy of making it to the other side of the monkey bars. What we will claim is that logic can be fun. Just ask any member of the United States Chess Federation.

      In fact, the logic we use in geometry is like a Vulcan kid's playground. It's reasonable, but not overly complex. It gets us piecing together logical statements and eventually building arguments in the form of proofs. And it's fun…for Vulcans, anyway.

    • Proofs

      The proof. The most petrifying aspect of geometry.

      It may instigate fear, anxiety, and terrible flashbacks of when Barry Meanbottom, the second-grade class jerk, said you didn't know how to swim. Not wanting the other kids to laugh, you of course openly declared that you could. "Prove it," Barry had said with a sneer. Even as you jumped that diving board you knew you were doomed. To this day, you still have anxiety attacks around proofs and swimming pools.

      Well, unlike Barry Meanbottom, geometry won't ever ask you to prove something you don't have the skills to prove. Sure, it might make you analyze what you already know and synthesize new information to make the proof work, but it's not a jerk. It'll make sure you have all the tools to swim and not sink. (That's a metaphor. We can't do anything about your actual swimming skills.)

      And unlike Barry's father, big-time corporate insurance lawyer Mr. Meanbottom, geometry won't ever make you prove something that isn't true. The proofs you'll write won't be fabricated evidence presented to a gullible (and probably tampered) jury.

      Instead, you'll work through factual proofs using logical arguments and arrive at reasonable conclusions that could convince any judge and more importantly, your math teacher.

      The bottom line is that proofs aren't as bad as most people think they are, and all because geometry is nothing like the Meanbottoms.

      Here's a video on one kind of proof.

    • Shapes


      When your gym teacher tells all you "lazy daisies" to "get in shape," he might want to be a bit more specific. Which shape is he talking about, exactly? A rectangle or a circle? A two-dimensional or three-dimensional one? Gym teachers aren't really known for their specificity, especially if the phrase "drop and give me twenty" is any indication.

      When you think of geometry, you can't help but think of shapes. Sure, we need logic and proofs to help us out along the way, but ultimately, 2D and 3D figures get to the heart of what geometry is all about: shapes we can see with our own two eyes.

      We're not picky about which shapes we learn about, either. We'll take any shape, from triangles (with 3 sides) to dodecagons (with 12 sides). Actually they don't even have to have sides at all, like circles! We'll even open it up to three-dimensional solids like prisms and cones.

      Geometry is about understanding the world around us, and our world takes the shape of…well…shapes. It's about understanding what makes shapes so special, how they're similar or different, and why. We'll learn about the inner workings of isosceles triangles, what makes trapezoids tick, and why spheres are so…well-rounded.

    • Dimensions


      Unlike the adventures of Meg Murry, our exploration of dimensions will max out at three. Any past that and we're in physics and/or sci-fi territory.

      We'll start with the first dimension: points and lines. There isn't that much to say about the first dimension, since all we can do is find locations and lengths. We have no widths or depths to work with, and number lines are interesting only up until a certain point.

      Don't worry. We'll speed past the first dimension pretty quick and jump right into the second, filled with lengths and widths, angles, lines, polygons, and circles. The majority of our time in Geometryland will be spent in the second dimension, exploring the relationships between shapes, angles, perimeters, and even areas.

      If everything is in 3D nowadays, why even bother with 2D? What's so special about it? Well, there are many reasons to investigate the second dimension.

      1. It's cool.
      2. Even though the real world exists in 3D, our sense of vision is limited to two dimensions only.
      3. The second dimension is the basis for the third one, so it's important to understand the ins and outs of the 2D world before we add depth to the mix.

      Once we've gotten a firm grip on what the second dimension can offer us, we can feel free to put on those 3D glasses and venture into the realm of nets, surface area, and volume.

      These are the only dimensions we need to capture the essence of geometry. Who says math isn't as easy as 1, 2, 3?

    • Constructions

      Knowing all the different properties of lines and shapes is important. But if we're truly going to rule the world of geometry, we need to be the absolute overlords of it. We're talking about a level of omnipotence that makes the very universe quake with fear.

      We need to know how to create constructions.

      It might sound like we're hammering up drywall and fixing the plumbing, but knowing how to construct shapes is a crucial skill in the world of geometry. How else can you create, change, and destroy geometric shapes at will?

      Constructions are helpful, too, especially when proofs seem to be going nowhere. The simple addition of a perpendicular or angle bisector might change a frustrating triangle problem into a much simpler proof than you may have realized. Cutting things in half often does that.

      You don't need any fancy rulers, protractors, or high-tech computer algorithms to make these shapes, either. Just pick up an old-school straightedge and compass, and the geometric world is your oyster-shaped construction.

  • Logic


    When you were a wee little child, your parents would take you to the neighborhood park to play. You'd swing and slide and climb on the jungle gym. Once in a blue moon, you brought colored chalk with you and played tic-tac-toe with your friends on the sidewalk. That was the closest you ever got to merging logic and fun.

    We aren't going to claim that the logic in geometry is as fun as a playground. Very few things in life beat the sheer joy of making it to the other side of the monkey bars. What we will claim is that logic can be fun. Just ask any member of the United States Chess Federation.

    In fact, the logic we use in geometry is like a Vulcan kid's playground. It's reasonable, but not overly complex. It gets us piecing together logical statements and eventually building arguments in the form of proofs. And it's fun…for Vulcans, anyway.

  • Proofs

    The proof. The most petrifying aspect of geometry.

    It may instigate fear, anxiety, and terrible flashbacks of when Barry Meanbottom, the second-grade class jerk, said you didn't know how to swim. Not wanting the other kids to laugh, you of course openly declared that you could. "Prove it," Barry had said with a sneer. Even as you jumped that diving board you knew you were doomed. To this day, you still have anxiety attacks around proofs and swimming pools.

    Well, unlike Barry Meanbottom, geometry won't ever ask you to prove something you don't have the skills to prove. Sure, it might make you analyze what you already know and synthesize new information to make the proof work, but it's not a jerk. It'll make sure you have all the tools to swim and not sink. (That's a metaphor. We can't do anything about your actual swimming skills.)

    And unlike Barry's father, big-time corporate insurance lawyer Mr. Meanbottom, geometry won't ever make you prove something that isn't true. The proofs you'll write won't be fabricated evidence presented to a gullible (and probably tampered) jury.

    Instead, you'll work through factual proofs using logical arguments and arrive at reasonable conclusions that could convince any judge and more importantly, your math teacher.

    The bottom line is that proofs aren't as bad as most people think they are, and all because geometry is nothing like the Meanbottoms.

    Here's a video on one kind of proof.

  • Shapes


    When your gym teacher tells all you "lazy daisies" to "get in shape," he might want to be a bit more specific. Which shape is he talking about, exactly? A rectangle or a circle? A two-dimensional or three-dimensional one? Gym teachers aren't really known for their specificity, especially if the phrase "drop and give me twenty" is any indication.

    When you think of geometry, you can't help but think of shapes. Sure, we need logic and proofs to help us out along the way, but ultimately, 2D and 3D figures get to the heart of what geometry is all about: shapes we can see with our own two eyes.

    We're not picky about which shapes we learn about, either. We'll take any shape, from triangles (with 3 sides) to dodecagons (with 12 sides). Actually they don't even have to have sides at all, like circles! We'll even open it up to three-dimensional solids like prisms and cones.

    Geometry is about understanding the world around us, and our world takes the shape of…well…shapes. It's about understanding what makes shapes so special, how they're similar or different, and why. We'll learn about the inner workings of isosceles triangles, what makes trapezoids tick, and why spheres are so…well-rounded.

  • Dimensions


    Unlike the adventures of Meg Murry, our exploration of dimensions will max out at three. Any past that and we're in physics and/or sci-fi territory.

    We'll start with the first dimension: points and lines. There isn't that much to say about the first dimension, since all we can do is find locations and lengths. We have no widths or depths to work with, and number lines are interesting only up until a certain point.

    Don't worry. We'll speed past the first dimension pretty quick and jump right into the second, filled with lengths and widths, angles, lines, polygons, and circles. The majority of our time in Geometryland will be spent in the second dimension, exploring the relationships between shapes, angles, perimeters, and even areas.

    If everything is in 3D nowadays, why even bother with 2D? What's so special about it? Well, there are many reasons to investigate the second dimension.

    1. It's cool.
    2. Even though the real world exists in 3D, our sense of vision is limited to two dimensions only.
    3. The second dimension is the basis for the third one, so it's important to understand the ins and outs of the 2D world before we add depth to the mix.

    Once we've gotten a firm grip on what the second dimension can offer us, we can feel free to put on those 3D glasses and venture into the realm of nets, surface area, and volume.

    These are the only dimensions we need to capture the essence of geometry. Who says math isn't as easy as 1, 2, 3?

  • Constructions

    Knowing all the different properties of lines and shapes is important. But if we're truly going to rule the world of geometry, we need to be the absolute overlords of it. We're talking about a level of omnipotence that makes the very universe quake with fear.

    We need to know how to create constructions.

    It might sound like we're hammering up drywall and fixing the plumbing, but knowing how to construct shapes is a crucial skill in the world of geometry. How else can you create, change, and destroy geometric shapes at will?

    Constructions are helpful, too, especially when proofs seem to be going nowhere. The simple addition of a perpendicular or angle bisector might change a frustrating triangle problem into a much simpler proof than you may have realized. Cutting things in half often does that.

    You don't need any fancy rulers, protractors, or high-tech computer algorithms to make these shapes, either. Just pick up an old-school straightedge and compass, and the geometric world is your oyster-shaped construction.

  • Key Skills


    If you're going to study geometry (and if you're reading this, chances are you probably will), then you might want to keep a few tricks of the trade in your back pocket. Take them out and use them whenever you need to.

    Some of those tricks might get a little uncomfortable if you sit on them for too long, so maybe your back pocket isn't the best place for them after all.

    • Logic and Reasoning


      You've gotten this far in math, so you're probably at least somewhat logical. Maybe you play chess or dabble in philosophy or know that riding your stationary bike to school isn't really an option. You make some sense, although your parents might beg to differ.

      That's a wonderful starting place, but geometry kicks it up a notch. It asks you to appreciate the intricacy and artfulness of a well-reasoned argument—and even come up with your own.

      And no, trying to convince your parents that tomato sauce makes pizza a vegetable doesn't count as a well-reasoned argument. Just because Congress agreed to it doesn't mean your folks will.

    • Algebra

      On the first day of school, you walked into geometry class with a smile on your face. Not because you like math or geometry, but just because you knew that algebra was truly done and over with. No more quadratic equations or polynomials or solving for x.

      Unfortunately, that only lasted for a good two minutes. You quickly realized that geometry, as different as it might seem, actually relies heavily on algebraic concepts. If you don't know how to work with exponents, simplify expressions, or solve for that aggravating x, you might end up hating geometry just as much as algebra.

      Now, we're not saying you'll need to remember every tiny detail about algebra. It's all right to refresh your memory. In fact, we highly encourage you to go back and look at your notes just to wipe the cobwebs off those concepts.

      If you and your friends doused them in gasoline and set them ablaze in a ritualistic bonfire at the end of last year, don't worry. You can borrow ours, free of charge.

  • Logic and Reasoning


    You've gotten this far in math, so you're probably at least somewhat logical. Maybe you play chess or dabble in philosophy or know that riding your stationary bike to school isn't really an option. You make some sense, although your parents might beg to differ.

    That's a wonderful starting place, but geometry kicks it up a notch. It asks you to appreciate the intricacy and artfulness of a well-reasoned argument—and even come up with your own.

    And no, trying to convince your parents that tomato sauce makes pizza a vegetable doesn't count as a well-reasoned argument. Just because Congress agreed to it doesn't mean your folks will.

  • Algebra

    On the first day of school, you walked into geometry class with a smile on your face. Not because you like math or geometry, but just because you knew that algebra was truly done and over with. No more quadratic equations or polynomials or solving for x.

    Unfortunately, that only lasted for a good two minutes. You quickly realized that geometry, as different as it might seem, actually relies heavily on algebraic concepts. If you don't know how to work with exponents, simplify expressions, or solve for that aggravating x, you might end up hating geometry just as much as algebra.

    Now, we're not saying you'll need to remember every tiny detail about algebra. It's all right to refresh your memory. In fact, we highly encourage you to go back and look at your notes just to wipe the cobwebs off those concepts.

    If you and your friends doused them in gasoline and set them ablaze in a ritualistic bonfire at the end of last year, don't worry. You can borrow ours, free of charge.