1. Chapter 10 Measurement Section 10.6 Volume

2. The volume of a three-dimensional shape is a measure of how much space it fills up or sometimes we say how much capacity it has. Examples of volume measure would be how much milk does a container of milk hold? In the same way that area is the number of non-overlapping squares that fill up a two-dimensional shape the volume is the number of cubes that fill up a three dimensional shape. The units that are used to measure this are called cubic units i.e. cubic feet, cubic inches, cubic meters etc. The right rectangular prism to the right will have a volume of 15 of the yellow cubic units. Volume of a Right Rectangular Prism The volume of a right rectangular prism can be found by multiplying together the lengths of it three dimensions: l, w and h. Volume = l ·w·h l w h

3. The volume of the right rectangular prism pictured to the right is found the following way. Volume = 4·5·2 = 40 cubic units 4 5 2 Volume of Any Right Prism (or cylinder) The volume of any right prism can be found by finding the area of one of the identical ends (the book calls this number B) then multiplying that number B by the perpendicular distance between the identical ends (the book calls this number h). Volume = (area of one end)·(distance between ends) = B·h h Area B To find the volume of the right triangular prism pictured to the right. Step 1. B = Area of right triangle end = ½·4·9 = 18 Step 2. Volume = B·h = 18·10 = 180 10 9 4

4. To find the volume of the cylinder pictured to the right. Step 1. B = area of circular end =  ·3 2 = 9  Step 2. Volume = B·h = 9  ·7 = 63  3 7 The buildings we were drawing earlier finding the volume is a matter of determining how many cubes are in the building. Normally we assume each cube is a cubic unit. A. What is the volume of the building pictured to the right if each cube is a cubic unit? Volume = 10 cubic units (There are 10 cubes.) B. What if each cube is 8 cubic units? Volume = 10·8 = 80 cubic units. Volume of a Pyramid (or Cone) The volume of a pyramid or cone is sort of like that of a prism, but you multiply by ⅓. (B is the area of the base.) Volume = ⅓ B·h Area of B h

5. To find the volume of the rectangular pyramid pictured to the right that has a rectangular base. Step 1. B = area of rectangular end = 4·5 = 20 Step 2. Volume = ⅓·20·12 = 80 4 5 12 In the example above the height distance is given. It is the perpendicular distance from the base to the top (tip) of the pyramid. If the distance along the slanted side is given you need to apply the Pythagorean Theorem to get the perpendicular distance. Step 1. B = area of rectangular end = 8·12 = 96 Step 2. Find height h First find the distance to the center of the base x. Second find distance from the center of the base to the top h. Step 3. Volume = ⅓·96· 8 12 10 8 12 x h 10