Volume of a Cone | Math with Mr. J

[Music] welcome to math with mr j [Music] in this video i'm going to cover how to find the volume of a cone and remember volume is the amount of space that a 3d figure or object takes up now when it comes to cones we can use the formula 1 3 times the area of the base times the height that's the formula at the top left of the screen but since the base of a cone is a circle we can input the formula for the area of a 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 third times the area of the circular base pi r squared times the height now we'll talk about why we multiply by one third after we finish number one knowing why we do this will give us a better understanding of the formula so let's jump into number one where we have a cone with a base radius of 5 inches and then a height of 12 inches and the first thing that we need to do is write out our formula so volume equals 1 3 times the area of our base so pi r squared since we have a circular base and then times the height now i wrote out the formula a little different than how it's written at the top of the screen that's okay because i'm still multiplying 1 3 times the area of the base times the height it doesn't matter how you represent the multiplication symbols or parentheses as long as you're multiplying one-third times the area of the base times the height so once we have our formula written we can plug in our radius and height so we have volume equals 1 3 times pi our radius of 5 squared times the height which is 12. now at this point we're ready to solve so we can plug that into a calculator and we'll have the volume of that cylinder i'm going to break it down or simplify it a little further though until we get it in terms of pi which means a number times pi then i'll calculate so the first thing that we need to do is 5 squared so volume equals 1 3 times well 5 squared means 5 times 5 that gives us 25 times pi 25 pi times 12. now i 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 a number in front of pi when multiplying a number by pi so we are ready to move forward here and we have a 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 times 12. i'm going to do 25 times 12 and that gives us volume equals one-third times well 25 times 12 is 300. and pi so one-third times 300 pi now we have one-third times 300 or one-third 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 times pi and i'm going to use the pi button on a 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 a calculator so again i'm going to use the pi button and we get volume equals 314 and i'm going to round the decimal to the hundredths place so we get sixteen hundredths so three hundred fourteen and sixteen hundredths or point sixteen and this is cubic inches because we're talking volume so that's our final answer right there so i want to explain why we multiply everything by one third if we take a cone and we put it in a 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 a cylinder with the same exact radius and height that cone is going to be exactly one-third the volume of that cylinder pretty cool how it works out actually now when we find the volume of a cylinder we use the formula the area of the base times the height so it looks really similar to this and this the only thing we add in is the one-third because again that cone is exactly one-third the size or volume of that cylinder let's move on to number two where we have a cone with a given diameter for the base of 14 centimeters 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 centimeters and then our height is 8 centimeters the first thing we do write out our formula so volume equals 1 3 times pi r squared times the height once we have that we plug in so volume equals 1 3 times 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 times the height of eight now we're ready to simplify this and break this down until we get our final volume so next we'll do seven squared so volume equals one-third well seven squared that means 7 times 7 and gives us 49. so we have 49 pi that's the area of the base in terms of pi and then multiplied by the height of 8. our next step is going to be 49 times 8. we don't have anything compatible with that one-third that we can cross cancel so 49 times 8 is our best option so we'll end up with volume equals one-third times 392 pi now we have to multiply one-third by 392. one-third and 392 are not compatible meaning they don't work out nicely like number one where we had one-third times 300 and got a 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 one-third times 392 is going to give us 392 over three so volume equals 392 over 3 pi and that's our answer in terms of pi but i want to back up and show you how i got 392 over three if you're unsure so again we did one third and i'll try to squeeze this in over here times 392. now 392 is a 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 one times 392 gives us our numerator of 392 and then we do our denominators so 3 times 1 gives us a denominator of 3. so again that's our answer in terms of pi so 392 thirds pi now we're going to calculate this and put it in decimal form by doing 392 thirds times pi now i'm going to use the pi button on a 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 410 and 50 hundredths i'm rounding the decimal to the nearest hundredth and this is centimeters cubed so again our volume 410 and 50 hundredths centimeters cubed so there you have it there's how you find the volume of a cone i hope that helped thanks so much for watching until next time peace