How to Find Volume of a Composite Figure

in this video we're going to talk about composite shapes this is great video to watch after you've seen the other videos that are on the shapes by themselves in this case we're going to be looking at combination shapes so here's the example that's on your screen composite shape consists of hemisphere and right cone both radius 28 centimeters the height of the cone is 84 Centimeters calculate the volume of the composite shape and work out the total surface area of the shape so there's two things that we have to do the first one is the volume so what we need to do is kind of organize our thoughts let's go ahead and say that we want to find the volume of the cone and we're going to add that to the volume of the hemisphere so we're going to start off by identifying what would be the volume of the cone so if you were to look at your formula pocket or if you were to find the formula online you're going to find that the formula for the volume of cone is 1 3 pi squared times the height and then the volume of sphere is going to be four thirds Pi radius cubed but we don't want the entire sphere we only want half of the sphere so we're going to go ahead and multiply this by one half because we only want half of it so now I've gone ahead and set up my formula so I'm going to substitute it's going to equal 1 3 pi times the radius which is 28 squared times the height of the cone remember that here in the diagram 84 Centimeters is the height of the comb because that is perpendicular to the center so that it's going to be 84 Plus 4 over 6 or 2 3 when you simplify this pi and again the radius is 28 cubed now all of this is going to go straight into your calculator so let's go ahead and multiply in the calculator what is 1 3 times pi times 28 squared times 84. so this is going to give me 68 964.2419 Etc Plus the second part which is the volume of the hemisphere is going to give me 45 976.16129 go ahead and add those two numbers together and we get grand total of one hundred fourteen thousand nine hundred and forty if you round to the nearest whole number cubic centimeters remember the volume is always cubic so now we have been able to figure out the answer to part let's go ahead now and move on to Part and working with Part we're going to write it out very similarly we're looking in this case for total surface area so total surface area is pretend that we're going to wrap this shape in either aluminum foil or we're going to wrap it with wrapping paper so it's going to be all around we need to figure out what is the surface area of the cone remember this is composite shape so this cone right now when find the surface area don't need the bottom Circle normally cone is the curved surface which is like the birthday hat and then at the bottom you have circle so in this case to find the total surface area I'm going to find the surface area of the curved surface of the cone and then I'm going to add it to the surface area of the hemisphere so this is what our goal for this problem is to figure out the surface area of the curved part of the cone I'm going to use the formula pi RL where stands for the slant height don't know the slant height here have in red right triangle because this is the actual height of the cone here is the slant height which is the hypotenuse of this triangle and the 28 is the radius so will be able to find sand height through Pythagorean theorem we're going to hold that for second we're going to hold that thought plus then the surface area of the hemisphere is going to be well the surface area of sphere is 4 pi squared but don't want the whole entire sphere only want half of it because it's the hemisphere and so I'm going to go ahead and like said earlier we're going to have to use the Pythagorean theorem I'm going to do it on the side 28 squared plus 84 squared is equal to squared take your calculator find 28 squared add that to 84 squared and that gives you sum of 7588 is equal to the slant height squared take the square root of that answer and we get approximately 87.1 so this is my slant height it's approximately 87.1 so in the formula I'm going to write down pi times the radius which is 28 times the slant height which is 87.1 plus one half of 4 is 2 so you can either leave one half times four or I'm just going to write 2 because know that one half of 4 is 2. pi and then the radius is 28 squared so in your calculator you should have the entire value for 87.1 it's 87.1091269615 Etc leave it there in the calculator so when you do your calculations you don't have round off error I'm going to go ahead and multiply that by pi and by 28 to get that very first part and that gives me 7662.519 plus now let's go ahead and multiply 2 times pi times 28 squared and we get 4926.01728 let's go ahead and add those two numbers together for grand total of 12 588.54 square centimeters and it all depends on what you need to round to if you're doing this for the IB exam then you would want to round two three significant figures which will be 12 600 square centimeters hope you found this video helpful remember to subscribe to my channel for more help with math so that you can say yes can do math with confidence until next time thanks for watching