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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 B - Right Triangles and Trigonometry

This unit is all about right triangles, and we hate to break it to you, but it's pretty important. Yeah, the Pythagorean Theorem is essential, and knowing the properties of special right triangles will certainly make your life easier, but none of that compares to what happens after: this unit is where you'll first tackle trigonometry. (Gasp!)

9 Geometry—Semester B - Quadrilaterals and Polygons

Quadrilaterals are shapes that have four sides. Seems simple enough, right? Well, not exactly. You might feel comfortable around squares and rectangles, but how much do you know about trapezoids? They're serious wildcards. We'll learn about all the different properties and proofs concerning quadrilaterals, and a few other polygons might sneak their way into the quadrilateral party. We can't blame them. It really is hip to be square.

10 Geometry—Semester B - Circles

We'll start by defining the circle and talking about a few of its properties (did you know it owns a mansion in Connecticut and beach house in Boca?). After a little bit about central angles, arc measures, and arc lengths, we'll learn about the equations of circles on the coordinate plane and show circles some love by performing a few constructions with them.

11 Geometry—Semester B - Area

As you might have guessed, this unit is all about area. Be ready for a return of triangles, squares, quadrilaterals, circles, and basically every shape we've ever talked about. Once we're finished, don't be surprised if you find yourself secretly calculating areas of soccer fields or road signs in your head. It happens to the best of us.

12 Geometry—Semester B - Three-Dimensional Geometry

How much area covers a simple space figure's surface? We've got you covered. How much stuff fits inside a three dimensional solid? We can figure that out, too. In this unit, we'll examine space figures inside and out. We'll stick to simpler solids, such as cubes, prisms, cylinders, and cones. Finding the volume of a tuba is not something we'd like to try right away. Trust us.

13 Geometry—Semester B - Statistics and Probability

We're back in the land of probability and statistics, here to pick up some more pearls of wisdom. In this unit, we'll explore frequency tables, "and" vs. "or" probability, and dependent events, as well as factorials, combinations, and permutations. We're hoping that we'll find enough of pearls of wisdom to make a pearl of wisdom necklace, but we'll settle for learning a thing or two instead.

Recommended prerequisites:

  • Geometry—Semester A

    • Sample Lesson - Introduction

    Lesson 11.04: Areas of Regular Polygons

    A photo of an old faded stop sign.
    Collaborate and listen.

    (Source)

    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

    1. Which of the following is the correct formula to find the measure of the exterior angle of a regular polygon if n equals the number of sides?

    2. Which of the following is the correct formula to find the measure of the interior angle of a regular polygon if n equals the number of sides?

    3. Which of the following is the correct formula for the area of a regular of polygon if r is the length of the apothem, s is the length of each side; and n is the number of sides?

    4. Which of the following accurately describes the apothem of a regular polygon?

    5. What is the measure of a single exterior angle of a square?

    6. What is the measure of a single exterior angle of a regular pentagon? Explain your reasoning.

    7. The measure of a single exterior angle of a regular polygon is 120°. What is the shape of the polygon?

    8. The measure of a single exterior angle of a regular polygon is 40°. What is the shape of the polygon?

    9. What is the measure of a single interior angle of a regular pentagon?

    10. What is the measure of a single interior angle of a regular octagon?

    11. What is the measure of a single interior angle of a regular 30-sided polygon? Explain your reasoning.

    12. The measure of a single interior angle of a regular polygon is 90°. How many sides does the polygon have?

    13. The measure of a single interior angle of a regular polygon is 140°. How many sides does the polygon have? Explain your reasoning.

    14. A regular hexagon has an apothem of length 3√3 and sides of length 6. What is the area of the hexagon to the nearest tenth?

    15. A regular polygon has an area of 100 square centimeters, an apothem of 5 centimeters, and a side length of 10 cm. What is the shape of this polygon?

    16. A regular polygon has an area of 24√3 square inches, an apothem of 2√3 centimeters, and a side length of 4 inches. What is the shape of this polygon? Explain your reasoning.

    17. An octagon with an apothem of 19.3 centimeters has an area of 1235.2 square centimeters. Find the octagon’s perimeter to the nearest tenth of a centimeter.