Volume of a Cone Math with Mr J

Welcome to math with Mr. In this video, I'm going to cover how to find the volume of cone. And remember, volume is the amount of space that 3D figure or object takes up. Now, when it comes to cones, we can use the formula 1/3 * the area of the base * the height. That's the formula at the top left of the screen. But since the base of cone is circle, we can input the formula for the area of circle into our volume formula to make it more specific. So, the formula to the right at the top of the screen. We have 1/3 * the area of the circular base, pi squared, * the height. Now, we'll talk about why we multiply by 1/3 after we finish number one. Knowing why we do this will give us better understanding of the formula. So, let's jump into number one, where we have cone with base radius of 5 in and then height of 12 in. And the first thing that we need to do is write out our formula. So, volume equals 1/3 * the area of our base, so pi squared since we have circular base, and then * the height. Now, wrote out the formula little different than how it's written at the top of the screen. That's okay because I'm still multiplying 1/3 * the area of the base * the height. It doesn't matter how you represent the multiplication symbols or parentheses as long as you're multiplying 1/3 * the area of the base * the height. So, once we have our formula written, we can plug in our radius and height. So, we have volume equals 1/3 * pi our radius of 5 squared * the height which is 12. Now, at this point, we're ready to solve. So, we can plug that into calculator and we'll have the volume of that cylinder. I'm going to break it down or simplify it little further though until we get it in terms of pi, which means number * pi. Then I'll calculate. So, the first thing that we need to do is 5 squared. So, volume equals 1/3 * Well, 5 squared means 5 * 5. That gives us 25 * pi, 25 pi. * 12. Now, put the 25 in front of the pi symbol because we simplified the area of the base and now we have it in terms of pi. And typically speaking, you'll put number in front of pi when multiplying number by pi. So, we are ready to move forward here and we have couple of different ways we can do the next step. So, one way, we can cross cancel or multiply 1/3 and 12, or we can do 25 * 12. I'm going to do 25 * 12 and that gives us volume equals 1/3 * Well, 25 * 12 is 300. And pi. So, 1/3 * 300 pi. Now, we have 1/3 * 300 or 1/3 of 300. That will give us 100 pi. And that's our answer in terms of pi. Now, I'm going to calculate the exact volume by multiplying 100 * pi. And I'm going to use the pi button on calculator. Now, it's common practice to also use the rounded version of pi, 3.14. Both are correct, but know that answers will vary slightly depending on if you use the rounded version, 3.14, or the pi button on calculator. So again, I'm going to use the pi button and we get volume equals and I'm going to round the decimal to the hundredths place. So, we get 1600ths. So, 314 and 1600ths or .16, and this is cubic inches because we're talking volume. So, that's our final answer right there. So, want to explain why we multiply everything by 1/3. If we take cone and we put it in cylinder with the same exact height and radius, let me try to draw this so we can picture it. So, if we put this cone in cylinder with the same exact radius and height that cone is going to be exactly 1/3 the volume of that cylinder. Pretty cool how it works out actually. Now, when we find the volume of cylinder, we use the formula the area of the base * the height. So, it looks really similar to this and this. The only thing we add in is the 1/3 because again, that cone is exactly 1/3 the size or volume of that cylinder. Let's move on to number two, where we have cone with given diameter for the base of 14 cm. We don't want the diameter though, we want the radius. And remember, the radius is half the diameter. So, our radius will be 7 cm and then our height is 8 cm. The first thing we do, write out our formula. So, volume equals 1/3 * pi squared * the height. Once we have that, we plug in. So, volume equals 1/3 * pi and remember, our radius is 7 because we're given the diameter and we need half of that for the radius. So, 7 squared * the height of 8. Now, we're ready to simplify this and break this down until we get our final volume. So, next we'll do 7 squared, so volume equals 1/3. Well, 7 squared, that means 7 * 7 and gives us 49. So, we have 49 pi. That's the area of the base in terms of pi. And then multiply by the height of 8. Our next step is going to be 49 * 8. We don't have anything compatible with that 1/3 that we can cross cancel. So, 49 * 8 is our best option. So, we'll end up with volume equals 1/3 * pi. Now, we have to multiply 1/3 by 392. 1/3 and 392 are not compatible, meaning they don't work out nicely like number one, where we had 1/3 * 300 and got clean answer of 100. So, what we can do in order to simplify this to in terms of pi is to put it in fractional form. 1/3 * 392 is going to give us 392 over 3. So, volume equals over 3 pi. And that's our answer in terms of pi. But want to back up and show you how got 392 over 3 if you're unsure. So again, we did 1/3 and I'll try to squeeze this in over here * Now, 392 is whole number, so we can put it over one to put it in fractional form. Now, we can multiply. So, in order to multiply fractions, we multiply straight across. So, 1 * 392 gives us our numerator of 392 and then we do our denominators. So, 3 * 1 gives us denominator of 3. So again, that's our answer in terms of pi. So, 392/3 pi. Now, we're going to calculate this and put it in decimal form by doing 392/3 * pi. Now, I'm going to use the pi button on calculator. If you're using the approximate or rounded version of pi, 3.14, your answer is going to be slightly different than mine. So, once we plug this in, we will get an answer of and 5000ths. I'm rounding the decimal to the nearest hundredth. And this is centimeters cubed. So again our volume 410 and 5000ths centimeters cubed. So, there you have it. There's how you find the volume of cone. hope that helped. Thanks so much for watching. Until next time. Peace.