Geometry 11 3 b Volume of Cones

volume of cones 11.3 there's seven previous videos for chapter 11 that are in the description in the geometry playlist if you need them in this chapter we've been learning about the volume of prisms and cylinders this is for the volume of cones if it has base area and height our formula is for volume equals one-third times the base times the height it's the same formula for pyramids we can also do the volume is equal to third pi squared and this pi squared would be the area of the circle times the height the base times the height we can find the volume of cone with radius 5 centimeters and height 12 centimeters so remember the radius is half the diameter so it's from the center point of the base to this side using the formula we would do for radius squared we would do 5 squared and that would be times height 12. that would give us 25 times 12 which is 300 so we have 1 3 pi times 300 that would be 100 pi centimeters cubed which would be approximately 314.15 and so on we could round this to the nearest tenth as and two-tenths centimeters cubed we can find the volume of cone with base circumference of 21 centimeters and height that is three centimeters less than twice the radius should be two times the radius minus three we use the circumference to find the radius circumference is equal to two pi two times pi times the radius that means we can divide both sides by 2 pi when we substitute in 21 pi for the circumference this cancels out as 1 and we get the radius is equal to 10.5 centimeters the second thing we do is use the radius to find the height so our height was equal to two times the radius minus three we do two times ten point five we subtract the three and we get eighteen centimeters for our height the third thing we do is use the radius and height to find the volume so we use our volume formula that the volume is equal to one-third times pi times the radius squared times height we know that the radius is now 10.5 centimeters and we know that our height is 18 centimeters we do the radius squared 10.5 times 10.5 we get 110.25 we multiply it by the height so here we have this equation we can cancel this three out and reduce it by the three there's six threes in 18 so that's reduced to six and our fraction is gone now we have pi times 110.25 times six that gives us 661.5 pi centimeters cubed which with our calculators is approximately 2078.2 centimeters cubed so to be little more accurate don't multiply it by 3.14 on your calculator hit the pi key all right now take look at this diagram we can see that we have length of 25 feet here and we've got radius of 7 feet if this cone were to be standing up on its base here that would be its slant height wouldn't it but it's also right triangle isn't it so that would be the hypotenuse of the right triangle so we can use the pythagorean theorem to find the height we have seven squared plus our height squared is going to equal 25 squared so to find this that would be like the squared in the pythagorean theorem wouldn't it and this 25 feet would be and this 7 feet would be we would get 7 times 7 is 49 plus squared is equal to 625 we subtract 49 from each side of the equation and we get that the height squared is equal to 576. we remove the two exponent by putting radical sign around this side and the square root of 576 is 24 so our height is 24 feet that we know the height we can find the volume we use the radius of 7 feet and the height of 24 feet to find the volume using our formula for the volume of cone we put in 7 for our radius so we have 7 squared which is 49 we can reduce using the 3. our fraction is gone and that becomes an 8 49 times 8 is 392 so we have 392 pi feet cubed which using our calculators is approximately 1231.5 feet cubed we can explore the effects of changing dimensions the length width and height of rectangular pyramid are multiplied by 1 4 we can describe the effect on the volume so our original dimensions this 24 length this 20 length this 20 inch height we would have volume of one-third 24 times 20 times 20 that's one-third times 9600 which is 3200 inches cubed if we multiply it by 4 we would multiply each of the measures by fourth times one fourth is six twenty times one fourth is five and again we have five for our height we do six times five times five which is one hundred fifty times one third is fifty inches cubed so using our original dimensions our volume was three thousand two hundred inches cubed when we multiplied it by one fourth our volume became fifty inches cubed and fifty is equal to one sixty fourth times three thousand two hundred if the length width and height of rectangular pyramid are multiplied by 1 the volume is multiplied by 1 4 cubed 1 4 times 1 4 times one-fourth is one-sixty-fourth this means if we have the volume for the original dimensions we can just multiply it by one-sixty-fourth to know what the volume would be if it was multiplied by 1 4. we can find the volume of 3d composite figure that contains cone and cylinder and round to the nearest tenth the volume of the cylinder is we can see the diameter is six centimeters the radius is half the diameter so our radius is going to be three for the formula we would do radius squared so we're going to do three squared that's nine the height of our cylinder is five centimeters so we're going to multiply it by the height five nine times five is 45 pi centimeters cubed for the volume of this cylinder now we can do the cone we would put our information in for the radius squared we have 3 squared again because it's got the same radius as the cylinder and if this is 10 centimeters and this is 5 centimeters then this must be 5 centimeters to make it 10. my drawing is not in proportion but using these numbers we would have three squared times five which would be nine times five we can cancel out this three and that nine as three there's three threes and nine so now we just have pi times 3 times 5 which is 15 pi centimeters cubed we found the volume of the cylinder is 45 pi centimeters cubed and the volume of the cone is 15 pi centimeters cubed and the volume of the composite figure is the sum of the volumes it would be 45 pi plus 15 pi which gives us 60 pi centimeters cubed which on our calculators is approximately 188.5 centimeters cubed so we had cylinder with cone on top and we found that it was approximately 188.5 centimeters cubed but what is the volume of the composite figure if the cone is inverted into the cylinder well instead of using the sum of the figures we would find their difference we knew the cylinder was 45 pi and the cone was 15 pi if they had the same diameter same radius same height as this one it's just the cone is now inverted we would do 45 pi minus 15 pi which is 30 pi centimeters cubed which is approximately 94.2 centimeters cubed the last part of the lesson 11.3 we're going to discuss cube roots then we're going to get into chapter 12 which is all about circles i'm going to start it off with 12.1 and talk about chord secant tangent lines and segments that intersect circles remember to write your formulas down in your notes where you can find them easily hope you're doing well and i'll see you for the last part of lesson 11.3 bye you