Geometry—Semester B
We've got all the right angles.
- Credit Recovery Enabled
- Course Length: 18 weeks
- Course Number: 310
- Course Type: Basic
- Category:
- Math
- High School
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Shmoop's Geometry course has been granted a-g certification, which means it has met the rigorous iNACOL Standards for Quality Online Courses and will now be honored as part of the requirements for admission into the University of California system.
This course has also been certified by Quality Matters, a trusted quality assurance organization that provides course review services to certify the quality of online and blended courses.
Newsflash: geometry is about more than simple little doodles. It's about huge complicated ones.
Lucky for you, those doodles are everywhere. From your spherical baby brother to your dad (who's a total square), shapes are how we make sense of the world around us. If that's the case, knowing how to spot 'em and show how they work could prove useful. At least that way, we'll be one step closer to avoiding a shapeless alien invasion.
With activities, problem sets, and solvable examples galore, we'll tackle just about everything. In Geometry Semester B, we'll
- start with right triangles and learn about the trigonometric ratios that come with them. (Don't worry. They don't bite.)
- box some rectangles, fend off polygons, and make our way around circles.
- learn about area and volume, which seem like simple concepts, but take up more space than you might think (and not just in your noggin).
- finish up with some statistics and probability. There's more to it than tossing cubes or flipping cylinders.
P.S. Geometry is a two-semester course. You're looking at Semester B, but you can check out Semester A here.
Unit Breakdown
8 Geometry—Semester A - Reasoning and Proof
Ever wanted to be like the detectives or lawyers on Law & Order? If so, then this unit is for you. We'll learn everything about logic and reasoning, from conditionals and contrapositives to syllogism and detachment. And of course, we can't forget the meat and potatoes of geometry: proofs. Pass the gravy, please.
9 Geometry—Semester A - Transformations
In this unit, we'll be the masters of motion, transforming images to wherever we want, facing whichever way we want, and even in whatever size we want. We can even decide whether we want to perform these transformations on or off the coordinate plane! Don't let the power get to your head, though.
10 Geometry—Semester A - Parallel and Perpendicular Lines
Soon after learning about the properties of parallel lines, we'll shake things up by throwing transversals into the mix. Make sure you hold onto your congruent angles because these angles and their theorems are the main course of this unit. We'll perform constructions, prove theorems, and even talk about the slopes of these lines on the coordinate plane. And naturally, leaving out perpendicular lines just wouldn't be right.
11 Geometry—Semester A - Congruent Triangles
It's the same ol' same ol'. Well, we think that sameness—or congruence—can be good. (Oreos haven't changed in over 100 years, and they're still delicious!) In this unit, we will learn how to prove congruence in triangles and how to use that congruence to solve problems. We won't help you find a better place to stash your Oreos, though. You'll have to do that on your own.
12 Geometry—Semester A - Relationships Within Triangles
Up to now, most of our lessons have been focused on the tasty outer shell of triangles—properties that deal with their sides and angles, and proving congruencies. That's all good and fine, but the time has come for us to dive into the creamy center—special line segments and centers within the triangle, just dripping with delicious new postulates, corollaries, and theorems for us to enjoy.
13 Geometry—Semester A - Similarity
You probably thought you had escaped triangles, but we aren't finished with them yet. We're going to cover some new triangle theorems. At this point, these will practically seem like second nature. We'll also learn about the scale factor, which is the ratio by which the polygon is reduced or expanded. Who knew so much information was bundled up in just three sides?
Recommended prerequisites:
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Sample Lesson - Introduction
Lesson 11.04: Areas of Regular Polygons
Not to get all political or anything, but if regular polygons had a political stance, it would be for equality. Equality for what? Sides and angles, of course. If they could get around a little better, you'd see them at the polls handing out flyers.
Regular polygons are those shapes with equal sides and equal angles. They can have anywhere from 3 sides to an infinite number of sides. If there were 1 million sides, they would each probably be really small.
Road signs love regular polygons. A stop sign is one of the many regular polygons you might see out and about; they are octagons with equal sides and angles. Yield signs, too. They are equilateral triangles. That's right; we've been working with these sneaky shapes all along. Equality, and regular polygons, are clearly where it's at.
Sample Lesson - Reading
Reading 11.11.04: Areas of Regular Polygons
With any polygon, whether regular or not, one way to find its area is to cut it into triangles and rectangles. That way, you can just find the area of the triangles and rectangles, and add 'em all up to find the area of the original polygon. We already know how to find the area of triangles and rectangles, so that makes it easier to find the area of new, more intimidating shapes.
This method works because of the Area Addition Postulate: if two shapes don't overlap, then their total area is equal to the sum of the areas of the individual shapes. That seems a bit "no duh" to us, but some mathematician had to go ahead and prove it. Not us, though; we're just going to move on.
If we want to find the area of a regular polygon, but we want to use a general formula instead of cutting it up into little pieces (because that just seems cruel), we need to know a new term. An apothem is the distance from the center of a regular polygon to the center of one of its sides. This is generally labeled as r, because it's also the radius of a circle drawn inside the polygon that only touches the sides.
That red line? Totally an apothem. But notice that we also split that regular polygon into triangles. (Seem familiar?) Each side of the polygon is the length of each triangle's base, and the apothem is the height. We smell a formula coming on. We'll call the length of each side of the polygon s, and there are n triangles total.
By adding the areas of all the triangles together, we get the formula for the area of a regular polygon:
A = (1/2)nsr
So really, this formula gives us an easy-to-use application of the good old Area Addition Postulate, without us having to cut up a polygon each time we want to find its area. Pretty convenient. It wouldn't hurt at this point to remember what interior and exterior angles are and how we can use them.
Recap
With any polygon, one way to find its area is to cut it into triangles and rectangles, find the area of the triangles and rectangles, and add up the areas to find the area of the original polygon.
This method works because of the Area Addition Postulate: if two shapes don't overlap, then their total area is equal to the sum of the areas of the individual shapes.
Here's the lowdown on the area of a regular polygon:
- r = length of apothem, the distance from the center to the middle of a side
- s = length of side
- n = number of sides
- A = (1/2)nsr
If you need to track down the interior angles of a polygon, remember that the sum of all the interior angles is (n – 2) × 180°. The angles of a regular polygon are all equal, so we can find the size of the individual angles by dividing the sum by the number of sides: (n – 2) × 180° / n.
The exterior angles are even easier to work with, because they always add up to 360°. An exterior angle will just be 360° / n.
Sample Lesson - Activity
- Credit Recovery Enabled
- Course Length: 18 weeks
- Course Number: 310
- Course Type: Basic
- Category:
- Math
- High School
Schools and Districts: We offer customized programs that won't break the bank. Get a quote.