Geometry 12 5 Volume of Pyramids and Cones

hey welcome back to geometry this is chapter 12 section 5 we're looking at the volume of pyramids and cones okay we talked about the volume of prisms previously and we found that the area was the base times the height the and that was the area of the base the volume how much fits in there is the area of the base times the height now we have pyramids now this pyramid has the same base area so these are the same area of base same size base just this is pyramid okay we want to figure out how does this volume compared to this one notice they also have the same height okay so can place this in here and it fits in there perfectly has the same height and has the same base area okay now to figure out the volume can fill this one up and then can dump that here see how much of the volume is that now if you were to look at that you can see you can see that okay it's below the halfway mark okay that's about third and actually it should be if we've done it exactly it should be exactly 1/3 so the volume of pyramid is one-third that of the prism with the same base area and the same height so we have pyramid with volume of one third base times the height the same goes for an oblique pyramid where we have the same area and the same height those would have the same volume now what about cylinder and cone now again we have cylinder here and cone that have the same base area and the same height same height so can fit that in there if it's exactly inside there there we go if it's in there too the height has the same base area okay so the cylinder relates to the the the cylinder relates to the cone or the cone to the cylinder in the same way that pyramid relates to prism and we can have look at that okay and again here if we look at this air notice that's little below halfway that is also third so same relation here the cone relates to the cylinder in the same way that pyramid relates to prison it is one-third one-third of the base times the height and since the base is circle that's PI squared times the height and the same thing goes for an oblique cone so here's an example if they're going to be mean to you and instead of giving you the radius or giving you the diameter they put the they put an angle in there and then I'll walk you through what we're gonna do to get around this if know this angle down here and know the height and here's right angle then can use sohcahtoa so now remember the volume is 1/3 hi r-squared times the height while don't know the radius so I'm going to try to find that so can find that using the tangent since know opposite and adjacent the opposite is the 11 the adjacent is the radius so I'm going to do tangent 58 opposite adjacent now have to solve this for so I'm gonna multiply by and then I'm gonna divide by tangent 58 and now can simplify that type that into your calculator that's approximately six point nine for the radius now that know the radius can plug it in for the radius in my formula and the height is 11 and then work that out we get approximately five hundred and forty eight point four inches cubed and that's all the fun we're going to have today