Math Antics - Area - Free Educational videos for Students in K-12

Math Antics - Area - By Mathantics

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00:03	Uh huh Hi , welcome to Math Antics . And
00:08	our last geometry video , we learned that all two
00:10	dimensional shapes have a one dimensional quantity called perimeter ,
00:14	which is basically the outline of the shape . In
00:17	this video . We're going to learn that these shapes
00:19	also have a two dimensional quantity called area . To
00:23	help you understand what area is . Let's start by
00:26	imagining a line that's one centimeter long . Now let's
00:30	imagine moving that line in a perpendicular direction , a
00:33	distance of one centimeter . But while we move it
00:36	, it leaves a trail almost like the end of
00:39	a paintbrush by moving the one dimensional line . That
00:42	way we formed a two dimensional shape and all of
00:46	the space or surface that we covered along the way
00:49	is the area of that shape , which as you
00:51	can see here is just a square . Okay ,
00:55	but how much area does this square have ? Well
00:58	, our original line was one centimeter long and we
01:01	moved at a distance of one centimeter . So we
01:04	could say that this shape is a square centimeter just
01:07	like a centimeter is a basic unit for measuring length
01:10	. A square centimeter is a basic unit for measuring
01:13	area . But there's other units for area too ,
01:16	for example , instead of a centimeter . What if
01:19	our line had been a meter long ? And what
01:21	if we had moved it ? one m ? The
01:23	area we'd have gotten would be one square meter or
01:28	what if our line was a mile long and we
01:30	moved in a mile . We'd have a square mile
01:32	of area . So just like with perimeter , the
01:35	units of measurement are very important when we're talking about
01:39	area . All right , so that gives you a
01:41	good idea of what area is . But how do
01:43	we calculate area mathematically ? Well , there's some special
01:47	math formulas or equations that we can use to find
01:50	the area of different shapes . In this video ,
01:53	we're going to learn the formula for squares and rectangles
01:56	and the formula for triangles to find the area of
02:00	any square or rectangle . All we have to do
02:03	is multiply its two side dimensions together . They're usually
02:06	called the length and the width . So the formula
02:10	looks like this area equals length times width . But
02:14	it's often written with just the first letters of each
02:17	word as abbreviations , day for area , L .
02:20	For length and W for width . So let's see
02:24	if that formula works for our original square centimeter .
02:27	If we multiply the length one centimeter times the width
02:31	. One centimeter . What do we get ? Well
02:33	one times one is just one . But what about
02:37	centimeter time centimeter centimeter time centimeter just gives us square
02:42	centimeters which we can write like this using a two
02:46	as an ex moment . We read this as centimeters
02:49	squared and it's just a short way of writing centimeter
02:53	time centimeter . So whenever you see units like centimeters
02:57	squared or inches squared or meters squared or miles squared
03:01	, you know it's a measurement of the two dimensional
03:03	quantity area . Okay , our formula area equals length
03:08	times width . Work for our square . Now let's
03:11	see if it works for a rectangle . Here's a
03:13	rectangle that's four cm wide and two cm long or
03:17	tall . First we plugged the length and width into
03:21	our formula two cm and four cm . Then we
03:25	just multiply two times four equals eight and centimeter time
03:29	centimeter is centimeters squared . So according to our formula
03:33	, the area of this rectangle is eight cm squared
03:37	and we can see that's correct . If we bring
03:39	back our original square centimeter , if we make copies
03:42	of it you can see that exactly eight of those
03:45	square centimeters would be the same area as this rectangle
03:49	. Great , let's try our formula on one more
03:51	rectangle . This rectangle is two cm long , but
03:55	only half a centimeter . What ? And our formula
03:58	area equals length times with tells us that we just
04:01	need to multiply these two sides together to get our
04:03	area . two has 1/2 equals one . So this
04:08	rectangle is also one square centimeter . How could that
04:12	be a square centimeter ? It's not even a square
04:15	. Ah But just because the shape takes up one
04:18	square centimeter of area , that doesn't mean it has
04:21	to be a square shape . It just means that
04:23	the total area would be equal or the same as
04:26	a square centimeter . You can see that if we
04:29	break up the rectangle in half and rearrange it ,
04:31	then it would form a square . In fact ,
04:35	we can use square units like square centimeters to measure
04:38	any area no matter what the shape is , it
04:41	could be a rectangle , a triangle , a circle
04:44	, or any other two dimensional shape . Okay ,
04:47	now that you know how to find the area of
04:49	any square or rectangle . Using our formula , we're
04:52	going to learn the formula for finding the area of
04:54	any triangle . But to do that , we're going
04:56	to start with a rectangle again . What the dimensions
05:00	of this rectangle are three m by four m .
05:03	So what's its area ? Well , using our formula
05:07	, we know that the area would be three m
05:09	times four m , Which is 12 m squared .
05:13	But now what if we were to kick this rectangle
05:16	exactly in half along a diagonal line from opposite corners
05:21	? It forms two triangles . And because each of
05:24	these triangles is exactly half of the rectangle , that
05:28	means that the area of either triangle must be exactly
05:32	half the area of the rectangle . We already calculated
05:36	that the area of the entire rectangle is 12 m
05:39	squared . So the area of this triangle must be
05:42	six m squared , and the area of this triangle
05:45	must be six m squared , since six is half
05:47	of 12 . Ha , ha . So the formula
05:51	for the area of a triangle should just be half
05:53	of a rectangle . So does that mean that instead
05:56	of area equals length times width , it should be
05:59	area equals one half of length times width , yep
06:03	. That's basically it . But with one important difference
06:06	instead of L . For length and w for width
06:09	, we're going to use two different names for our
06:11	triangles dimensions , we're gonna call them base and height
06:15	. And here's why the names length and width worked
06:18	okay for this right triangle , because the right triangle
06:21	is exactly half of a rectangle . But those names
06:24	don't really work for other kinds of triangles , like
06:27	acute triangles or up to strangles because how do you
06:30	tell which side should be which ? So for triangles
06:34	we do something different . First we choose one of
06:37	the three sides to be the base , it doesn't
06:40	really matter which side you choose . And in a
06:42	lot of math problems , the base will already be
06:44	chosen for you . Once we decide which side the
06:47	bases , we imagine setting the triangle down on the
06:50	ground so that its base is flat on the ground
06:53	. Like this . Next we find the highest point
06:57	of the triangle which is the vertex that's not touching
07:00	the ground . From that point we draw a line
07:03	straight down to the ground . The line we draw
07:06	must be perpendicular with the ground . The length of
07:09	that line from the tip of the triangle to the
07:12	ground is called the height of the triangle . Oh
07:15	some people call the height of a triangle the altitude
07:18	. Which makes a lot of sense . If you
07:20	pretend that you're triangle is a tiny little mountain .
07:28	Sometimes the height line is inside the triangle like with
07:31	an acute triangle . And sometimes it's outside the triangle
07:35	like with an obtuse triangle . And sometimes it lines
07:38	up exactly with one of the triangle sides , like
07:41	with right triangles . But no matter where it is
07:44	, the formula for finding the area of any triangle
07:47	is the same Area equals 1/2 base times height .
07:52	So if we know these two measurements , base and
07:55	height , we can just plug them into our formula
07:57	to calculate the area . At first . You might
08:00	not see how the same formula could work for all
08:03	three types of triangles . But watch this here's an
08:06	acute triangle and this box is one half its base
08:10	times its height . If we cut our triangle up
08:12	, you can see that it fits perfectly inside that
08:15	area . But wait , there's more , here's an
08:18	obtuse triangle with a box that's one half its base
08:21	times its height . Again , if we cut up
08:24	the triangle it fits perfectly inside the box . Now
08:28	you can see how the formula area equals one half
08:31	base times height works for any kind of triangle .
08:34	Okay , we already figured out that the area of
08:37	this right triangle was 6 m2 . So let's practice
08:41	using our new formula to calculate the area of these
08:44	last two triangles . Our Diagram shows that the base
08:48	of this acute triangle is five m and its height
08:51	is eight m . So we plug those values into
08:54	our formula for area and we get area equals one
08:57	half of five times eight . Five times eight is
09:01	40 , and one half of 40 is 20 .
09:04	So the area of this triangle is 20 m squared
09:08	. Don't forget that the units of measurement for area
09:10	will always be square units . Okay , that was
09:14	pretty simple . Let's try our last example , the
09:17	Diagram of this up to strangle tells us that the
09:20	base is four and the height is 7" . So
09:24	let's plug those values into our formula . We end
09:27	up with the equation , area equals one half of
09:31	four times seven , four times 7 would be 28
09:34	. And then we can calculate what one half of
09:36	28 would be by dividing by two , 28 ,
09:39	divided by two is 14 . So the area of
09:43	this obtuse triangle must be 14 sq in . Okay
09:48	, now , you know all the basics of area
09:50	, you know that area is a two dimensional quantity
09:53	that we measure in square units . You've learned the
09:56	formula for calculating the area of any square or rectangle
10:00	area equals length times width . And you've learned the
10:03	formula for calculating the area of any triangle area equals
10:07	one half of base times height . Do , but
10:11	don't forget to practice what you've learned by working some
10:14	problems on your own . That's how you really get
10:16	good at Math as always . Thanks for watching Math
10:19	Antics and I'll see you next time . Learn more
10:23	at Math Antics dot com .

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