What is volume? Volume is a way of measuring space. For example, how much space is in a rectangular room that has floor measurements of 12 ft. by 16 ft. and a wall or height measurement of 12 ft. To measure space we use a cube 1 ft. by 1 ft. by 1 ft. or 1 cubic foot. So, how many of these cubes will it take to fill the above room? 1 ft. 1 ft Click for sound
16 12 Click for sound We can stack 16 times 12 cubes on the floor or 192 cubes and then we can stack these 192 cubes 12 layers high for a total of 2304 cubes measured in feet, so we call it cubic feet. The volume of our room is 12 * 12* 16= 2304 cubic feet Click here for floor plan
A rectangular solid is what you might think of as a box shape. All the sides are perpendicular to each other and the three dimensions that it has (length, width, and height) may be different measurements. Formula for volume is V = lwh Click for sound l w h
A cube is a rectangular solid in which all of the sides are equal in length. Formula for volume is V = e 3 where e is the measure of a side. Click for sound e e e e
A sphere is what you would think of as a ball, no sharp edges, round all over. Formula for the volume of a sphere is where r is the radius of the sphere. Click for sound
A cylinder is what we might think of as a can. While we may have in mathematics slanted cans, the ones in the store are what we call a right circular cylinder in that the sides are perpendicular to the horizontal. The base and top of the can is a circle and thus has a radius r, the distance between the top and bottom is called the height of the can or h. If cut and straightened out this shape would be a rectangle. Click for sound h r Area of top is
A right circular cone is similar to an ice cream cone. In mathematics there are slanted cones, but for our purposes we will be looking at the right circular cone, whose base (which is a circle) is perpendicular to the horizontal. r h R is the radius at the base of the cone. H is the height of the cone. The formula for the volume is Click for sound
Use the formulas to compute the volume of the objects in the following problems. When necessary round your answers to the nearest hundredth. When writing your final answer, use the appropriate units, i.e. cu ft. A new convention for writing cubic units or square units is to use an exponent on the type of unit, for example; cubic feet would be written ft 3. When finished check your answers on the last page. 1. Rectangular solid: L = 73mm, W = 17.2 mm, H = 16 mm (mm is millimeters) 2. Cube: e = 17.3 in (inches)
3. Sphere: r = 8.2 in 4. Sphere: diameter= 76.4 cm 5. Cylinder: r = 13.5 in, h = 8.2 in 6. Cylinder: d = 16.2 m, h = 7.5 m In = inches, cm = centimeters, m = meters d= diameter of circle = 2 r or 2 radii 7. Cone: r = 1.4 cm, h = 5 cm 8. Cone: d = 9.5 in, h = 7 in Work out these problems before going to the next slide.
3. Sphere: r = 8.2 in 4. Sphere: diameter= 76.4 cm 5. Cylinder: r = 13.5 in, h = 8.2 in 6. Cylinder: d = 16.2 m, h = 7.5 m 7. Cone: r = 1.4 cm, h = 5 cm 8. Cone: d = 9.5 in, h = 7 in 1. Rectangular solid: L = 73mm, W = 17.2 mm, H = 16 mm (mm is millimeters) 2. Cube: e = 17.3 in (inches) 20,089.6 mm 3 5177.72 in 3 2309.56 in 3 233,495.60 cm 3 4694.95 in 3 1545.90 m 3 10.26 cm 3 165.39 in 3
Congratulations! You have just completed the geometry unit.