Name: 数学アンチックス - エリア (Math Antics - Area)
Uploaded: 2021-01-14T06:24:02.000Z
Duration: 10 min 27 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

In our last geometry video, we learned that  all 2-dimensional shapes have a 1-dimensional quantity called Perimeter

様態の副詞

which is basically the outline of the shape.

In this video, we're going to learn that these shapes also have a 2-dimensional quantity called Area.

To help you understand what Area is, let's start by imagining a line that's 1 cm long.

Now, let's imagine moving that line in a perpendicular direction a distance of 1 cm.

But while we move it, it leaves a trail… almost like the end of a paint brush.

By moving the 1-dimensional line that way, we formed a 2-dimensional shape,

and all of the space (or surface) that we covered along the way is the Area of that shape.

…which as you can see here is just a square.

Ok, but how much area does this square have?

Well, our original line was 1 cm long, and we moved it a distance of 1 cm,

so we could say that this shape is a square centimeter.

Just like a centimeter is a basic unit for measuring length,

a square centimeter is a basic unit for measurement for area.

For example, instead of a centimeter, what if our line had been a meter long, and then we moved it 1 meter?

The area we'd have gotten would be 1 square meter!

Or, what if it was a mile long, and we moved it a mile!?  We'd have a square mile of area.

So, just like with perimeter, the units of measurement are very important when we're talking about area!

Alright, so that gives you a good idea of what area is, but how do we calculate area mathematically?

Well, there's some special math formulas (or equations) that we can use to find the area of different shapes.

In this video, we're going to learn the formula for squares and rectangles and the formula for triangles.

To find the area of any square or rectangle, all we have to do is multiply its two side dimensions together.

They're usually called the length and the width, so the formula looks like this: Area equals length times width.

But it's often written with just the first letters of each word as abbreviations:

‘A’ for Area, ‘L’ for Length and ‘W’ for Width.

So let's see if that formula works for our original square centimeter.

If we multiply the length (1 cm) times the width (1 cm), what do we get?

Well, 1 x 1 is just 1, but what about cm x cm?

Cm x cm just gives us square centimeters, which we can write like this using a '2' as an exponent.

We read this as "centimeters squared"  and it's just a shorter way of writing cm x cm.

So whenever you see units like centimeters squared, or inches squared, or meters squared, or miles squared,

you know it's a measurement of the 2-dimensional quantity area.

Ok, our formula (area equals length times width) worked for our square.

Now let's see if it works for a rectangle.

Here's a rectangle that's 4 cm wide  and 2 cm long (or tall)

First, we plug the length and width into our formula

the area of this rectangle is 8 centimeters squared.

And we can see that's correct if we bring back our original square centimeter.

you can see that exactly 8 of those square centimeters would be the same area as this rectangle.

Great, let's try our formula on one more rectangle.

This rectangle is 2 cm long but only half a centimeter wide.

And our formula (area equals length times width)

tells us that we just need to multiply those two sides together to get our area.

So this rectangle is also 1 square centimeter.

How can it be a square centimeter?   It's not even a square!

Ah - but just because a shape takes up 1 square centimeter of area,

that doesn’t mean it has to be a square shape.

It just means that the total area would be equal (or the same) as a square centimeter.

You can see that if we break the rectangle in half and rearrange it, then it would form a square.

In fact, we can use square units (like square centimeters) to measure ANY area,

It could be a rectangle, a triangle, a circle  or ANY other 2-dimensional shape.

Okay, now that you know how to find the area of any square or rectangle using our formula,

we're going to learn the formula for finding the area of any triangle.

But to do that, we're going to start with a rectangle again.

The dimensions of this rectangle are 3 m by 4 m.

Well, using our formula,  we know that the area would be 3 m x 4 m which is 12 meters squared.

But now, what if we were to cut this rectangle exactly in half along a diagonal line from opposite corners?

And because each of these triangles is exactly half of the rectangle,

that means that the area of either triangle must be exactly half of the area of the rectangle.

We already calculated that the area of the entire rectangle is 12 meters squared,

so the area of this triangle must be 6 meters squared,

and the area of this triangle must be 6 meters squared, since 6 is half of 12.

Ah ha!  So the formula for the area of a triangle should just be half of the rectangle.

So does that mean that instead of, "Area equals length times width" , it should be,

"Area equals one-half of length times width" ?

Instead of 'L' for Length and 'W' for Width, we're going to use two different names for our triangle's dimensions.

We're going to call them "Base" and "Height"  and here's why.

The names "Length" and "Width" worked okay for this right triangle

because a right triangle is exactly half of a rectangle.

But those names don't really work for other kinds of triangles like acute triangles or obtuse triangles.

Because how do you tell which side should be which?

So for triangles, we do something different.

First we choose one of the three sides to be the "Base".

It doesn't really matter which side you choose,

and in a lot of math problems, the base will already be chosen for you.

we imagine setting the triangle down on the ground so that its base is flat on the ground, like this…

Next, we find the highest point of the triangle, which is the vertex that's not touching the ground.

From that point, we draw a line straight down to the ground.

The line we draw must be perpendicular with the ground.

The length of that line (from the tip of the triangle to the ground) is called the "height" of the triangle.

Oh, and some people call the height of a triangle the "altitude"

which makes a lot of sense if you pretend that your triangle is a tiny little mountain.

Sometimes the height line is inside the triangle,

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数学アンチックス - エリア (Math Antics - Area)

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