Geometry: Volume - Prisms, Cylinders, Cones, Pyramids and Sphere
This is a free lesson from our course in Geometry
 
   
This section explains the concepts of Volume and combines important properties of Cube, Prism, Pyramids, Cylinders, Cones, and Spheres, to apply the relevant formulas for determining the volume of the 3D solids. As a first step, you’ll need to review earlier learning about the solids, surface areas and use the formulas for volume. Moving forward you’ll be able to calculate area of base, lateral area(s) and volume of these shapes. Further on, you’ll solve several problems involving the volumes3D solids. Once mastered the volume of cubes, prisms etc. can move to other 3D figures including pyramids, cones, and spheres etc. The presentation covering such content will be done by the instructor in own handwriting, watching video and with the help of several examples with solution. This will help you apply important geometry properties and relationships to solve problems from day to day life situations and also doing geometry home work. (More text below video...)
<h2> Geometry- Volume - Prisms, Cylinders, Cones, Pyramids and Spheres </h2> <p> volume, solids, prism, cylinder, cone, pyramid, formula, video, sphere, area, base, diameter, length, width, height, geometry, volume of solids, Volume: Prisms Cylinders Cones Pyramids and Spheres, example, solution, rectangular prism, formula, calculate, radius of sphere, practice questions, quizzes</p> <p> to find the volume of a rectangular prism with dimensions are 12 cm, 6 cm and 8 cm., substitute values of dimensions in the above formula</p>
Other useful lessons:
Surface Area - Prisms, Cylinders, Cones, Pyramids and Spheres
Effect of dimension changes on volume
(Continued from above) Cube: The measure of the occupied space is called the Volume of the object. In cases where the object is hollow then the interior is empty and can be filled in with air or some liquid then, then the volume of the substance that can fill the interior is called the Capacity of the container. Thus the unit of measurement in both the cases is cubic unit. If an object is solid, then the space occupied by such an object is measured and is called the Volume of the object. On the other hand if the object is hollow then the interior is empty and can be filled with air or some liquid then will take the shape of its container. In this case, the volume of the substance that can fill the interior is called the Capacity of the container. In short the volume of the object is the measure of the space it occupies and the capacity of an object is the volume of the substance that can be accommodated inside. Hence the unit of measurement of either of the two is cubic unit.
The unit of measurement of volume is unit cube. A unit cube is the volume of a cube having side as ‘one unit’. E.g. if the side of a cube is 1 cm, its volume is one cubic centimeter (cm3) and if the side is 1 m, the volume is 1 cubic meter (m3). In other words to find the volume of a cuboid, you are to determine the number of unit cubes contained in it.
 
To find the volume of a cuboid, explore to find the number of unit cubes contained in it. E.g. the number of unit cubes in the cuboid (Fig: 1) are, 5 × 4 × 2 = 40. So, the volume of the cuboid can be said to be 60 cm3.
The formula for Volume of a cuboid, as a general expression be written = length * breadth * height. Since cube has all the sides (edges) equal, therefore, Volume of cube = (edge)3.
Cube
 
Look at some of the examples to illustrate the concepts and use of above formulae:
Example 1: Given- find the volume of a wood plank measuring 4 m in length, 2.5 m in breadth and 20 cm in thickness.
Step 1: make sure that all the dimensions are in the same measurement units.
Step 2: Given- l = 4 m, b = 2.5 m and h = 20 cm = 20/100 m = 1/5 m (measure- 1m = 100cm)
Step 3: Volume = length × breadth × height = l * b * h
= 4 × 2.5 × 1/5 m3 = 2.0 m3.
The volume of cuboid is 2 m3, as the final answer.
 
Example 2: Given- The volume of water in the tank measures 21,000 M3. If the length and breadth of the tank are 60 m and 35 m respectively, find its depth.
Step 1: Given-
   volume of water in tank = 21,000 cu m
   length of tank = 60 m
   breadth of tank = 35 m
Step 2: assume depth of the tank is x m, Thus its volume = (60 * 35 * x) m3
Step 3: x * 60 * 35 = 21000.
Step 4: Simplifying it equals to x = 10
The depth of the tank is 10 m, as the final answer.
Example 3 : Given- you are given a 5 cm3 cube painted on all its side, and asked to slice it into 1 cm3 . How many 1 cm3 cubes will have exactly one of their sides painted?
Step 1: When 5 cm3 cube is sliced into 1 cm3 cubes, you will get 5 * 5 * 5 = 125 Nos. (1 cm3) cubes.
Step 2: Notice that on each side of the larger cube, the smaller cubes on the edges will have more than one side painted. Thus the cubes which are not on the edge of the larger cube and that lie on its facing sides will have exactly one side painted.
Step 3: The larger cube has, 5 * 5 = 25 cubes on each face. Out of this, 16 cubes will be on the edge and 9 cubes are not on the edge.
Step 4: As seen in step 3 above, 9 (1 cm3) cubes on each face will have painting on one of their sides.
Thus total number of such cubes will be, 9 * 6 = 54, as the final answer.
Example 4: Given- the dimensions of a cuboidal shape beam are (length 10 m, breadth 60 cm and thickness 25 cm). How much you will have to pay to get it, if it costs $250.00/ M3?
Step 1: Notice that all the measurements should be expressed in the same units (100 cm = 1 m).
Step 2: Volume of the beam = [length x breadth x height (thickness here)].
        = 10 * (60/100) * (25/100) = 1.5 m3
Step 3: Cost you will be required to pay = 1.5 * 250 = $375.00, as the final answer.
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