Study Guide

Algebraic Expressions - Geometric Formulas

Geometric Formulas

Formulas wouldn't tell us squat if we couldn't turn around and apply them to situations in the real world. Fortunately, we can do just that. Formulas can help us figure out how to deal with, plan for, or manipulate objects of all different shapes and sizes, both two-dimensional and three-dimensional. The two-dimensional ones are easier, as you might imagine, so let's start there. This ain't a throw-you-in-the-deep-end-of-the-pool-to-teach-you-to-swim kind of lesson.

Two-dimensional shapes show up a lot in the world. A soccer field is a rectangle, the body of a bicycle may be a triangle, and some cakes have circular tops. These are all three-dimensional objects, but with two-dimensional parts. Practical considerations aside, two-dimensional shapes are good for math problems because we can draw them on paper. Because most paper that we know of is more or less 2D.

We'll run through some familiar shapes in the next few pages. For each shape, we'll give you a formula for the perimeter, meaning the distance around the outside of the shape, and a formula for the area, meaning the size of the surface covered by the shape. Perimeter is measured in units of length such as inches, feet, or miles. Area is measured in units of length squared, such as square inches, square feet, or square miles. Perimeter usually works better in poetry. For example, Robert Frost never would have been such a hit if he'd written that he had "square miles to go before I sleep."

  • Four-Sided Shapes

    Don't you love it when you can rely on common sense rather than having to memorize complicated formulas? Yeah, us too. Well then, you'll absolutely adore four-sided shapes. You might even marry them. You and four-sided shapes, sitting in a tree, K-I-S-S-I-N-G.

    To determine the distance all the way around a four-sided shape, you simply add the lengths of the four sides. If you remember this step, you can figure out whatever formula you need by drawing a picture of the shape and thinking for 30 seconds. If that makes your brain throb, you can think for 15 seconds, take a quick snack break, then come back and think for 15 more.

    A parallelogram is a four-sided figure in which opposite sides are parallel. Opposite sides will be equal, and opposite angles will be equal as well. Equality reigns supreme here in Paralleloland. Man, that's a lot of L's.

    The perimeter is the sum of all four sides. We could write this as the formula P = 2b + 2where P stands for perimeter. However, it's more important (and simpler) to remember that the perimeter is the sum of all four sides than to commit this formula to memory. You've got enough numbers and symbols floating around in that noggin of yours. Let's free some space for a change.

    The area of a parallelogram is its height times its base. To see why that is, chop off the left corner of the parallelogram (watch your fingers):

    Then move it over to the right side:

    Now we have a rectangle that has the same area as the parallelogram. The area of this rectangle is bh, and therefore, the area of the parallelogram is also bh. What a copycat. If we let be the area of the parallelogram, we can express this fact via the formula:

    A = bh

    A rectangle is just a particular type of parallelogram: it has right angles at all four corners. Since a rectangle is a parallelogram, opposite sides will still have the same length. Note that this fact doesn't work in reverse—not all parallelograms are necessarily rectangles. Some of them are Virgos.

    The perimeter P of a rectangle is given by this formula:

    P = 2l + 2w

    And the area A of a rectangle is given by this formula:

    A = lw

    square is a rectangle with all four sides the same length. Squares live in very strict, cookie-cutter communities. Also, they're restricted in the number of guests they can have in their backyard on weeknights.

    The perimeter P of a square is:

    P = 4s

    A square's area is:

    A = s2

    A trapezoid is a four-sided figure with one pair of opposite sides parallel. The sides that are parallel to each other are called the bases of the trapezoid. That may be a little confusing, since we're used to thinking of a base as being only on the bottom of something, but this feller does indeed have a base on top as well. He's got all his bases covered.

    In the picture below, b1 and b2 are the bases.

    As with a square, the perimeter is the sum of all four sides. While we could write a formula for perimeter in terms of the names of the sides, it wouldn't make any more sense than writing "the perimeter is the sum of all four sides," so let's not go there. Formulas are cool and all, but straightforward concepts expressed in plain English are the bomb.

    Here's the area A of the trapezoid:

    To see the justification for this formula, read about the distributive property and simplifying expressions. However, since that'll involve a lot of clicking and reading, our hunch is that you'll assume we know what we're talking about. That's perfectly fine, too.

     
  • Three-Sided Shapes

     

    We call a three-sided shape a triangle. You know these guys; they've gotten pretty famous thanks to pizza.

    The perimeter of a triangle is the sum of its three sides. The area A of a triangle is given by this formula:

    This formula is the way it is because the triangle fills exactly half of a rectangle with side lengths b and h, as shown below.

     
  • Circles

    circle is the collection of all points at a given distance r from a specified point. If you've grown tired of collecting baseball cards or Wii games and want to take up a new hobby, start collecting points, and perhaps you too can one day be a circle. It's a lofty goal, but who are we to tell you it'll never happen? Dream on, dreamer.

     

    The specific point we're referring to is called the center of the circle, and the distance r is called the radius of the circle.

    Here's a fun—and yet, at the same time, incredibly frustrating—puzzle to consider: how many sides does a circle have? Depending on how "side" is defined, potential answers may vary. If a "side" must be a straight line, then a circle clearly has no sides. If a "side" can be curved, then a circle has one side. Perhaps a circle has infinitely many sides? Or maybe it has a "good side" and a "bad side." It does look a little more handsome when it turns to the left. The question, "How many sides does a circle have?" is basically too ambiguous to answer. We'll answer it with an equally ambiguous, "Hmm...eh."

    Since we can't add the side lengths of a circle to find its perimeter, we need a special formula and a fancy name to go along with it. The perimeter of a circle is called its circumference, and it's given by the formula:

    C = 2πr

    ...where C means circumference, and r means radius.

    The area A of a circle is given by A = πr2. And don't forget, π is an irrational number that comes out to about 3.14 when we round it.