 Hi there 13.2 is all about volume of pyramids and cones Take out your notes slide this into your table of contents and get ready for some high-quality note-taking experiences So first recall the volume of a prism was the area of one base times the height of the prism Likewise the volume of a cylinder. This is just review is pi r squared times height Well for pyramids and cones The volume is just one-third of their sort of related prisms and cylinders So a pyramid is One-third times the area of one base times the height of the pyramid Now remember height of the pyramid is that distance from the apex or the tippy top to the base and Likewise the volume of a cone one-third pi r squared times height So for example if we were to find the volume of this square pyramid We're using the volume is one-third base times height formula We know since we've got a square base that that B is 36 square units and Then the height well to find the height, which is the purple segment a g We're going to use the triangle made up of points a g and e Segment g e is the app of them And so with the square if you double up the apathome you have one full side length So half of a side length is three and so the apathome g e is three units long We're given a e the green length is five units and so the apathome Can be found using sorry not the apathome, but the height can be found using the Pythagorean theorem So now we have all we need to know Volume is one-third times 36 times 4 which is 48 cubic units of fun How about this thing we want to find the volume of that cone and remember well volume of a cone is one-third pi r squared h so Volume if you sub in those values volume is one-third times pi times 1.5 squared times 8 Now notice 1.5 squared is 2.25 and I can multiply in any order. I want so I'll multiply one-third times 2.25 times 8 and that gives me 6 I'll tack on the pi at the end to give me a final answer of 6 pi cubic units Here's another example We want to find the exact volume of a regular hexagonal pyramid the height of 8 and a base edge of 10 We know we're dealing with the volume formula one-third area the base times the height We're given the height of the prism. So that's nothing we need to find we already know it We do need to find however the area of the base Now we're given one base edge Now since all of the base edges are 10 if I split it up split up one base edge I get five and Then remember with hexagons that little half triangle is a 30 60 90 and so the apathome here is 5 root 3 and so the base area is 150 root 3 and Now I have everything I need to find the volume of the pyramid So volume is one-third times base, which is 150 root 3 times height, which is 8 and So I can multiply in any order. I choose I'm gonna multiply one-third times 150 times 8 first And then I'll tack on that root 3 at the end So I have an exact volume of 400 root 3 cubic units Another example here. We have we want to find the volume of a regular triangular pyramid the base edge of 12 and a height of 10 So the height isn't shown But we do know that h equals 10 and we need to find the area of one base Given one base edges 12 So since we have a regular triangle, that's another way of saying an equilateral triangle the area of the base We can use side squared root 3 over 4 to get a base area of 36 root 3 and So we substitute those values into the area of pardon me the volume formula one-third times 36 root 3 times 10 Which gives us a volume of 120 root 3 cubic units Here we want the approximate volume of this regular pentagonal pyramid The apathome is 3 the slant height is 5 so again, we're using the volume is one-third base times height formula and We need to find the base area remember the base is a regular pentagon The apathome is 3 and so using trigonometry. We can find half of one side length Which means if I double that up and multiply by five I'll get a perimeter of about 21 point seven nine six I'll use that perimeter then to find the area of the base Then I need to find the height of the pyramid now to find that height Using that right triangle that's made up of the apathome height and slant height But that green theorem says that that height of this pyramid must be four And so therefore I sub in the base area I found I sub in the height that I found and we get a volume of 43.6 cubic units Next hey, let's pretend a cone has a volume of four 480 pi cubic units Its height is 10. How big is its radius? Well, we know we're dealing with this formula because we've got a cone Then we know the volume and we know the height we need to solve for radius So well, I see both sides of the equation have pi so I'll divide by pi I've got that pesky one-third, so I'll multiply both sides of the equation by three and Then I'll divide by 10 and so then I get r squared is 144 Which means the radius must be 12 units Need the volume of this thing Now you'll probably recall that we once found the surface area of this thing And it was a little bit tedious to find the surface area Well finding the volume is a little bit more straightforward Because all we need to do is find the volume of the prism and the volume of the pyramid add them together So prism of course is base times height Pyramid is one-third base times height But the heights of the prism and the pyramid are different so I'll add in little subscripts h1 and h2 First I know the area of the base is just 10 squared easy enough The h1 which is the height of the prism is 14 Now the height of the pyramid is going to be a little bit trickier so I want to find that length and So I'll make myself a right triangle using the radius of the square this Pardon me the lateral edge, which is 13 and then the height So once again, we need the lateral edge We need to find the radius of the base So the radius of the base if the base is 10 Using special right triangles. I know that the radius is 5 root 2 and so now we can find the height Of the pyramid using the good old Pythagorean theorem And so now we have all the information we need for our calculation For this one since approximation is okay. I'm going to pull out the calculator and I get about 1763.6 cubic yards. All right last example. We want the exact volume of this thing What is this thing? Well, this thing is a cone and another cone stuck together. They have the same circular base But that circular base is kind of smushed together. So we've got cone plus other cone I'll use blue to represent the cone on the left and green for the cone on the right So I know I've got one-third pi r squared h plus one-third pi r squared h The radius in both of these cones is the same. It's 3 but the heights are different. So I have blue height green height The blue height on the left is just given to us. It's 4. The green height is unknown however Because we've got a right triangle made up of the radius height and slant height and That 30 degree angle. I know that the height of that cone off to the right is 3 root 3 And now since we want an exact volume, we're going to keep our pi We're going to keep our root 3 as much as possible So I've got one-third pi r squared h plus one-third pi r squared h And I know I can multiply in any order. I choose so I'm going to leave the pi at the end I'm also going to leave the root 3 at the end So one-third 9 times 4 is half sorry one-third of 36 which is 12 pi and one-third times 9 times 3 is 9 so we'd have 9 root 3 pi and Strange as it seems that is our final answer if we want an exact answer. Hey, thanks for watching guys