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...)
(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 (cm^{3}) and if the side is 1 m, the volume is 1 cubic meter (m^{3}). 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 cm^{3}.
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}.
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 m^{3} = 2.0 m^{3}.
The volume of cuboid is 2 m^{3}, as the final answer.
Example 2: Given
The volume of water in the tank measures 21,000 M^{3}. 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) m^{3}
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 cm^{3} cube painted on all its side, and asked to slice it into 1 cm^{3} . How many 1 cm^{3} cubes will have exactly one of their sides painted?
Step 1: When 5 cm^{3} cube is sliced into 1 cm^{3} cubes, you will get 5 * 5 * 5 = 125 Nos. (1 cm^{3}) 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 cm^{3}) 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/ M^{3}?
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 m^{3}
Step 3: Cost you will be required to pay = 1.5 * 250 = $375.00, as the final answer.
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