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Math 102: College Mathematics14 chapters | 108 lessons

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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.

How do you find out the area of rectangles and triangles? Learn how in this lesson! We'll look at the formulas, then practice solving problems for each shape.

Rectangles and triangles are all around us... watching us. That's not creepy at all, right? Okay, it's a little creepy. But, they're also around us in less ominous, more friendly ways.

That field where you play soccer or football? That's a rectangle. As is everything from your cell phone to the state of Colorado. And that rack you use to set up the balls on a pool table? That's a triangle. So is the peanut butter and jelly sandwich you cut diagonally because it's more fun to eat that way. They even come in larger sizes, like the pyramids in Egypt. In this lesson, we're going to learn how to find the area of these shapes. Let's start with rectangles.

Let's say you're planting a garden. If your garden is 12 feet long and 4 feet wide, how big is it? This garden is a rectangle, and what you want to know is its area. The **area of a rectangle** is the length times the width.

So, your garden is 12 feet long by 4 feet wide. 12 is our length, and 4 is our width. 12 * 4 is 48. So, your garden is 48 square feet. Note the 'square.' That means that there are 48 one foot by one foot squares in your garden. That's plenty of space for your vegetables!

Let's try another. Let's say you've planted your garden, and now you have an abundance of zucchini, so you're making zucchini bread - mmm, tasty. Your bread pan is 9 inches long by 5 inches wide. What is the area? Like your garden, your pan is rectangular. Ah, the circle of life. Or, rectangle of life.

So, we want the length times width, which is 9 * 5. That's 45. So, your pan is 45 square inches. Okay, now you have to deal with all your extra peppers, cucumbers, carrots and other vegetables. You had quite a harvest. You decide to can them for the winter. You keep the jars on the shelves in your apocalypse preparedness closet (you know, just in case). If a shelf can hold 6 jars from front to back and 10 jars from left to right, and you have 300 jars (I said it was a big harvest), how many shelves will you need?

So, our rectangular shelf is 6 jars wide by 10 jars long. 6 * 10 is 60, so one shelf holds 60 jars. That's our area. But wait, we have 300 jars. How many 60-jar shelves will we need for 300 jars? Just divide 300 by 60 to get 5. So, we'll need 5 shelves.

With a belly full of zucchini bread and all that canning behind us, let's talk triangles. Let's say your garden went so well you decide to branch out, so to speak, into growing and selling Christmas trees. That's ambitious of you.

You look at your remaining yard. It's 30 feet by 50 feet. So, the area of that rectangle is 30 * 50, or 1500 square feet. What if you cut it in half diagonally, like below?

Now you can grow trees in one half and still have half a yard for your annual croquet tournaments.

How big is the area of the triangle you have for trees? It's half the rectangle, which was 1500 square feet. So, it's 750 square feet. Do you know what you just did? You just figured out the area formula for triangles. The **area of a triangle** can be defined as 1/2 * base * height or just 1/2*bh*. Base and height are just like length and width in a rectangle. In the right triangle above, which is a triangle with a right angle, we have half a rectangle. So, it makes sense that our formula is the same as a rectangle but cut in half.

So, you plant your trees in your triangular plot and wait. Let's jump forward in time. It's a math lesson. We can do that. Your trees are now grown. Here's one below!

You notice it forms a triangle. Can you find the area of that triangle?

We know the base is 4 feet. But, it doesn't form a right triangle. That's okay! We call this an oblique triangle, which is just any triangle that doesn't have a right angle. To determine the area, we just need to know its height, which is the vertical red line above. It's the line that's perpendicular from the base. Perpendicular means it forms a right angle, which gets us back into what we had with our right triangle. That's 7 feet. Our formula doesn't change: 1/2 * base * height, or 1/2 * 4 * 7. That's 14. So, the area of the triangle is 14 square feet.

Staring at your trees makes you hungry, so you decide to eat some pizza. When you look at a slice, you realize that, like your fir trees, it's a triangle! Only your pizza slice is much tastier. If the slice is 5 inches along the crust and 7 inches from top to bottom, what's its area? Well, 5 inches is our base and that 7 inches is our height. So, this pizza slice's area is 1/2 * 5 * 7. That's 17.5 square inches. Okay, well, subtract a few for the bite you just took while doing the math.

In summary, you've proven to be quite the botanist. And, you've also mastered some essential geometry. We learned that the **area of a rectangle** is length times width. We just need to know those two sides, and we can determine the area.

With a triangle, it's 1/2 times base times height. In a right triangle, our base and height are the two sides that meet to form a right angle. In an oblique triangle, the height is just a line that's perpendicular from the base.

After this lesson, you should be able to:

- Identify the formulas for finding the area of a rectangle and a triangle
- Define right triangle and oblique triangle

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Math 102: College Mathematics14 chapters | 108 lessons

- Go to Logic

- Go to Sets

- Properties of Shapes: Rectangles, Squares and Rhombuses 5:46
- Properties of Shapes: Triangles 5:09
- Perimeter of Triangles and Rectangles 8:54
- Area of Triangles and Rectangles 5:43
- The Pythagorean Theorem: Practice and Application 7:33
- How to Identify Similar Triangles 7:23
- Applications of Similar Triangles 6:23
- Parallel, Perpendicular and Transverse Lines 6:06
- Types of Angles: Vertical, Corresponding, Alternate Interior & Others 10:28
- Angles and Triangles: Practice Problems 7:43
- Properties of Shapes: Circles 4:45
- Go to Geometry

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