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Properties of Shapes: Quadrilaterals, Parallelograms, Trapezoids, Polygons

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  • 0:06 Quadrilaterals
  • 1:53 Parallelograms
  • 3:07 Trapezoids
  • 4:50 Polygons
  • 6:01 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Four-sided objects are more than just squares and rectangles. In this lesson, we'll discuss quadrilaterals, parallelograms and trapezoids. We'll also discuss polygons, objects that can have more sides than you can count.

All About Shapes

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.

Quadrilaterals

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:

Squares, rectangles and boomerangs are all quadrilaterals.
square, rectangle and boomerang shapes

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.

The perimeter of this quadrilateral is 15 inches.
quadrilateral with perimeter of 15

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.

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.

Example of a parallelogram
parallelogram with perimeter 18

When we have a specific type of quadrilateral like the one below, we can determine its area.

The area of a parallelogram is base times height.
diagram of parallelogram

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:

The area of this parallelogram is 21 square inches.
area of parallelogram

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!

Trapezoids

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?

The perimeter of this trapezoid is 24 inches.
diagram of 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 area of this trapezoid is 28 square inches.
area of trapezoid

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.

Polygons

We've been talking about four-sided objects, but what's the term that also includes shapes with different numbers of sides?

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