Volume of a Cone | Math with Mr. J - Free Educational videos for Students in K-12

Volume of a Cone | Math with Mr. J - By Math with Mr. J

Transcript


00:0-1	Welcome to Math with mr J . In this video
00:05	, I'm going to cover how to find the volume
00:07	of a cone . And remember volume is the amount
00:10	of space that a three D . Figure or object
00:13	takes up . Now when it comes to cones we
00:16	can use the formula one third times the area of
00:20	the base times the height . That's the formula at
00:24	the top left of the screen . But since the
00:27	base of a cone is a circle we can input
00:30	the formula for the area of a circle into our
00:34	volume formula to make it more specific . So the
00:37	formula to the right um at the top of the
00:40	screen we have one third times the area of the
00:44	circular base pi r squared times the height . Now
00:49	we'll talk about why we multiply by one third .
00:51	After we finish number one , knowing why we do
00:55	this will give us a better understanding of the formula
00:59	. So let's jump in the number one where we
01:01	have a cone with a base radius of five inches
01:04	and then a height of 12 inches . And the
01:06	first thing that we need to do is write out
01:09	our formula . So volume equals one third times the
01:16	area of our base . So pi r squared since
01:19	we have a circular base and then times the height
01:22	. Now I wrote out the formula a little different
01:25	than how it's written at the top of the screen
01:27	. That's okay because I'm still multiplying one third times
01:32	the area of the base , times the height .
01:34	It doesn't matter how you represent the multiplication symbols or
01:38	parentheses . As long as you're multiplying One 3rd times
01:42	the area of the base , times the height .
01:46	So once we have our formula written we can plug
01:48	in our radius and height . So we have volume
01:53	equals 1 3rd times pi Our radius of five squared
02:04	times the height , which is 12 . Now at
02:09	this point we're ready to solve so we can plug
02:12	that into a calculator and we'll have the volume of
02:16	that cylinder . I'm going to break it down or
02:18	simplify it a little further though , until we get
02:21	it in terms of pie , which means a number
02:24	times pi , then I'll calculate . So the first
02:27	thing that we need to do is five squared ,
02:31	So volume equals 1 3rd times while five squared means
02:37	five times five . That gives us 25 times pi
02:41	25 pi times 12 . Now , I put the
02:48	25 in front of the pi symbol because we simplified
02:51	the area of the base and now we have it
02:53	in terms of pie . And typically speaking you'll put
02:57	a number in front of pie when multiplying a number
03:01	by pi . So we are ready to move forward
03:04	here and we have a couple of different ways we
03:07	can do the next step . So one way we
03:10	can cross cancel or multiply 1/3 and 12 or we
03:14	can do 25 times 12 . I'm going to do
03:17	25 times 12 and that gives us volume equals one
03:24	third Times while 25 times 12 is 300 and Pie
03:33	. So 1 3rd times 300 pie . Now we
03:38	have 1/3 times 300 or 1/3 of 300 . That
03:43	will give us 100 pie . And that's our answer
03:49	. In terms of pie . Now I'm going to
03:52	calculate the exact volume by multiplying 100 times pi and
03:57	I'm going to use the pie button on a calculator
04:00	. Now it's common practice to also use the rounded
04:03	version of Pi 3.14 . Both are correct , but
04:08	know that answers will vary slightly depending on if you
04:12	use the rounded version 3.14 or the pie button on
04:16	a calculator . So again I'm going to use the
04:19	pie button and we get volume equals 314 and I'm
04:25	going to round the decimal to the 100th place .
04:29	So we get 16/100 so 314 and 16/100 or 3140.16
04:37	and this is cubic inches because we're talking volume ,
04:42	so that's our final answer right there . So I
04:46	want to explain why we multiply everything by 1/3 .
04:52	If we take a cone and we put it in
04:54	a cylinder with the same exact height and radius ,
04:58	let me try to draw this so we can picture
05:00	it . So if we put this cone in a
05:04	cylinder with the same exact radius and height , that
05:10	cone is going to be exactly one third of the
05:14	volume of that cylinder . Pretty cool how it works
05:18	out actually . Now , when we find the volume
05:20	of a cylinder , we use the formula , the
05:23	area of the base times the height . So it
05:27	looks really similar to this and this . The only
05:31	thing we add in Is the one third because again
05:35	that cone is exactly 1/3 the size or volume of
05:40	that cylinder . Let's move on to number two .
05:43	Where we have a cone with a given diameter for
05:46	the base of 14 centimeters . We don't want the
05:49	diameter though . We want the radius . And remember
05:52	the radius is half the diameter , so our radius
05:56	will be seven centimetres and then our height is eight
06:00	centimeters . The first thing we do right out our
06:03	formula , so volume equals one third times pi r
06:11	squared times the height . Once we have that we
06:16	plug in so volume equals one third times pi .
06:23	And remember our radius is seven because we're given the
06:26	diameter and we need half of that for the radius
06:29	so seven squared Times the height of eight . Now
06:34	we're ready to simplify this and break this down until
06:37	we get our final volume . So next we'll do
06:41	seven square so volume equals one third . Well seven
06:46	squared that means seven times seven and gives us 49
06:50	. So we have 49 pi . That's the area
06:54	of the base in terms of pie And then multiply
06:57	by the height of eight . Our next step is
07:01	going to be 49 times eight . We don't have
07:04	anything compatible with that one third that we can cross
07:07	cancel . So 49 times eight is our best option
07:11	. So we'll end up with volume equals one third
07:17	times 392 Pi . Now we have to multiply one
07:25	third by 390 to 1 third and 392 are not
07:30	compatible , meaning they don't work out nicely like number
07:34	one . Where we had one third times 300 got
07:38	a clean answer of 100 . So what we can
07:41	do in order to simplify this to in terms of
07:44	pie , is to put it in fractional form .
07:47	One third times 392 is going to give us 392
07:53	over three . So volume equals 392 over three pie
08:03	. And that's our answer in terms of pie .
08:06	But I want to back up and show you how
08:08	I got 392 over three if you're unsure . So
08:13	again we did one third and I'll try to squeeze
08:16	this in over here , times 392 . Now 392
08:22	is a whole number so we can put it over
08:25	one to put it in fractional form . Now we
08:28	can multiply , so in order to multiply fractions ,
08:31	we multiply straight across . So one times 392 gives
08:36	us our numerator of 392 and then we do our
08:41	denominators . So three times one gives us a denominator
08:45	of three . So again that's our answer in terms
08:49	of pie . So 392/3 pie . Now we're going
08:54	to calculate this and put it in decimal form by
08:57	doing 392 3rd times pi Now I'm going to use
09:02	the pie button on a calculator . If you're using
09:05	the approximate or rounded version of pi 3.14 your answer
09:10	is going to be slightly different than mine . So
09:13	once we plug this in we will get an answer
09:16	of 410 and 50/100 . I'm rounding the decimal to
09:24	the nearest 100th and this is centimeters cubed . So
09:30	again our volume 410.50 cm cubed . So there you
09:39	have it . There is how you find the volume
09:41	of a cone . I hope that helped . Thanks
09:44	so much for watching until next time . Peace .
09:53	Yeah .

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