Back To CourseELM: CSU Math Study Guide
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Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
Triangles? They're ok, but you hear about them everywhere. Squares, too. What about the more unusual shapes? Those ones with the funny names and weird properties. Let's learn a bit about them.
First up: the quadrilateral. It's a mouthful, I know. But, what do you see in that word? Quad-. Where have you seen that? Quadruplets? How many babies is that? It's four! Or, about three more than a normal person can handle. A quadrilateral is just a four-sided shape.
Squares and rectangles are quadrilaterals. So are more unusual shapes, like the boomerang below:
What do we know about quadrilaterals? If they have four sides, they also have four vertices, or corners. And, while those vertices can be any angle, they need to add up to 360 degrees. Why? Think about a square. All the angles are 90 degrees.
90 times 4 is 360. Now, what about that boomerang?
If that first angle is 130 degrees and the two small corners are 20 degrees, when we make those last two lines meet, the angle has to be 190 degrees. 130 + 20 + 20 + 190 = 360. No matter how you change the angles, they always add up to 360.
The perimeter of any quadrilateral is just the sum of the four sides. In the quadrilateral below, the sides, in inches, are 4, 4, 5 and 2.
Therefore, the perimeter is 4 + 4 + 5 + 2, or 15 inches.
Is every square a quadrilateral? They always have exactly four sides, so yes. But, is every quadrilateral a square? Well, no. Some are rectangles. And, some are other things, like parallelograms.
Do you remember what parallel means? Two parallel lines are two lines that will never meet. They're like the lines painted on the sides of an endless straight road.
A parallelogram is a four-sided shape with opposite sides that are parallel. Because of the parallel lines, the opposites sides are equal in length.
There are some unique properties of the angles inside parallelograms. First, the opposite angles are equal. Also, the adjacent angles are supplementary, which means they add up to 180 degrees.
The perimeter of the parallelogram below, where the sides are 4, 4, 5 and 5, is 18.
When we have a specific type of quadrilateral like the one below, we can determine its area.
The area of a parallelogram is the base times the height. So, in this example, where the base is 7 inches and the height is - whoa, wait, 4 is not the height! Remember, the height is the distance from the base to the top, so while that side is 4 inches, the height is actually 3 inches:
So, the area is 7 times 3, or 21 square inches.
This is the same area formula as with squares and rectangles. And, if you remember, squares and rectangles also have sets of parallel sides, so they're both parallelograms!
But, what if only one set of sides of a quadrilateral is parallel? Then, it can't be a parallelogram. But, it is a trapezoid. A trapezoid is a four-sided shape with at least one set of parallel sides. It can have two and be a parallelogram. But, if two sides aren't parallel, then it's just the lowly trapezoid.
That's a weird name, and it can be kind of a weird shape. You know what it reminds me of? A table. And, guess what? That's where the name comes from! Like a table, two sides, or the tabletop and the base, or the floor, need to be parallel. If not, your poor meatball may roll off the table and right out the door.
So, in a trapezoid, the parallel sides are called the bases. And, just like with a table, the other sides are called legs.
The perimeter is the same as with parallelograms. The perimeter of this trapezoid?
Let's see. The bases are 8 inches and 6 inches. The legs are 5 inches each. 8 + 6 + 5 + 5 = 24 inches.
Area is a little more complicated. You can't just multiply base times height, because the bases are different. So, the area of a trapezoid is the average of the bases times the height. Or, (a + b) / 2 * h.
Let's find the area of this one:
The bases are 8 inches and 6 inches. What's the height? The legs are each 5 inches. Remember, the height is the distance from one base to the other. The height of this trapezoid is 4 inches. Let's plug those numbers into our formula: 8 + 6 = 14. 14 / 2 = 7. 7 * 4 = 28. That's 28 square inches.
We've been talking about four-sided objects, but what's the term that also includes shapes with different numbers of sides?
Whoa. No. That's polygamy, or marrying multiple people. I meant polygons. Totally different. But, notice that they both begin with poly-. The prefix poly- comes from the Greek word for many. And, -gon means angles. So, a polygon is a closed, two-dimensional shape with many sides and angles. Polygons are polygamous in that they are shapes that join multiple sides and angles.
There's really no more to them. They can have three sides or eight, like an octagon, or a million, like the megagon. They don't need to have parallel lines or right angles. If you think about that, it means that a triangle is a polygon. So are squares, rectangles and, yes, quadrilaterals, parallelograms and trapezoids. They all are closed shapes with many sides, so they're all polygons!
The perimeter of a polygon is still just the sum of the sides. It's 3 for the top shape below, 5 for the one on the left, and 8 for the shape on the right.
Since polygons can get awfully complicated, determining their areas is more advanced math than we're going to get into here. But, no matter how big or small that stop sign is, you still need to obey it!
In summary, a quadrilateral is any four-sided shape.
A parallelogram is a four-sided shape with opposite sides that are both parallel and equal in length. The area of a parallelogram is the base times the height.
A trapezoid is a four-sided shape with at least one set of parallel sides. The area of a trapezoid is the average of the bases times the height.
Finally, a polygon is a closed, two-dimensional shape with many sides. Everything from a triangle to an octagon to a megagon is a type of polygon.
For all of these shapes, the perimeter is the sum of the sides.
At the end of this lesson, you'll be able to:
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Back To CourseELM: CSU Math Study Guide
15 chapters | 136 lessons