Volume of Solids

In mathematics, volume is a crucial measuring quantity of three-dimensional figures, such as a cube, cuboid, sphere, etc, used to calculate how each figure occupies space. It is a cornerstone in understanding basic rules and real-life applications of three-dimensional geometry. So, let’s explore everything about solids, from their units to formulas involved for solving questions related to the volume of these figures. 

1.0Introduction to 3-Dimensional Figures and Their Volume

3-Dimensional Geometry involves the study of solid shapes and their length, width, and depth, making them able to hold space. Unlike the flat, 2D shapes which occupy an area, 3D solids take up real space. Volume is basically the measurement of this space, the material it can hold or the area it contains. Volume in mathematics changes in the formula from shape to shape according to the geometry of the solid; for example, the volume of a cone is completely different from that of a sphere or cuboid. 

2.0Volume Formula for 3D Shapes

As mentioned earlier, the volume formula for three-dimensional shapes differs from each other in calculation, based on their shape and geometry. Let’s discuss the most commonly used formulas for some basic 3D shapes: 

1. Volume of Cube: A cube is a perfect solid, meaning all the faces of a cube are equal squares. It has equal width, breadth, and length, so all the edges are equal. You can imagine a cube as a square pulled equally in all directions to form a box with perfect symmetry. Like squares in 2D geometry, all the corners of a cube are at right angles, and all the faces are the same. The formula for the volume of the cube for side “a” can be expressed as: 

$\text{The Volume of Cube}=a^3$

2. Volume of Cuboid: A cuboid resembles a stretched cube, a box-shaped solid with all faces being rectangles. Opposite faces are equal, and while all the angles are right angles, the edges may vary in length. It's basically a 3D rectangle with three different dimensions: length (l), breadth (b), and height (h). The volume of a cuboid is written as: 

3. Volume of Cylinder: A cylinder is a regular, tube-shaped solid with two identical circular ends on the top and bottom, joined together by a curved surface. Understand the shape of a cylinder as a circle being drawn upward; this provides the figure with height (h) and volume. The axis passes through the centre of both circular ends, and the surface curves around evenly. The volume formula for a cylinder is: 

$\text{The Volume of Cylinder}=\pi r^{2}h$

4. Volume of Cone: A cone has a flat circular base, which tapers smoothly to a single point, known as the apex. It's similar to a triangle that has been turned around one of its sides, so it has a round base and curved surface. Its height (h) is measured directly from the apex to the centre of the base, whereas the slant height measures the length down the side. The formula to calculate the volume of a cone is given as: 

5. Volume of Sphere: A sphere is the most symmetrical 3D figure, as every point on its surface is exactly the same distance from the centre. It is shaped like a perfectly round ball and has no edges or vertices, unlike other solids. The volume of a sphere is calculated using the formula: 

6. Volume of Hemisphere: A hemisphere is simply a half sphere. When you cut a sphere equally in the middle along its diameter, you will be left with two hemispheres. It contains a levelled circular face and a curved face. It retains the radius(r) of the original sphere, as well as having the same centre for the curved section. The formula for the volume of a hemisphere is: 

3.0Units of Volume

The volume of any solid figure is always calculated in cubic units, such as cm³, m³, Litre, millilitre, etc, as it is the measure of space in three dimensions. These units are interchangeable according to the requirement of the question or in a practical situation. For example: 

4.0Measuring the Volume of Solids

Follow the steps below to efficiently calculate the volume of different solids, using the various formulas mentioned earlier:

1. Identify the shape of the solid whose volume needs to be calculated, and use the volume formula accordingly. 
2. Convert the units of all the quantities of solids, such as length, breadth, height, or radius, into a single unit, if not provided beforehand. 
3. Now, simply substitute these converted values into the required formula and simplify the equation.  

5.0Solved Examples on Volume of Solids

Problem 1: A solid toy is made by attaching a hemisphere to the top of a cylinder. The radius of both the hemisphere and the cylinder is 4 cm. The height of the cylinder is 10 cm. Find the total volume of the toy in cubic centimetres.

Solution: Given that, 

Radius of hemisphere = Radius of Cylinder = r = 4cm 

Height of cylinder (h) = 10 cm

Total volume of the toy = Volume of hemisphere + volume of Cylinder 

Total volume of the toy = $\frac{2}{3}\pi r^3+\pi r^{2}h$

Total volume of the toy = $\pi r^2(\frac{2}{3}r+h)=\frac{22}{7}\times (4{)}^2\times (\frac{2}{3}\times 4+10)$

Total volume of the toy = $\frac{13376}{21}=636.96c{m}^3$

Problem 2: A solid metallic cube of side 12 cm is melted and recast into a solid sphere. Find the radius of the sphere formed.

Solution: Given that, 

The length of the side of the metallic cube (a) = 12 cm 

According to the question, 

Volume of the Cube = Volume of the Sphere

$a^3=\frac{4}{3}\pi r^3$

$12\times 12\times 12=\frac{4}{3}\times \frac{22}{7}\times r^3$

$r^3=\frac{3\times 12\times 12\times 7}{22}$

$r=\sqrt[3]{\frac{3024}{22}}=\sqrt[3]{412.36}\approx 7.45cm$

Problem 3: How many small cubes of side 2 cm can be cut from a cuboid of dimensions 10 cm × 6 cm × 4 cm?

Solution: Given that the side of the required cube (a) = 2cm 

Let the number of cubes that can be cut out from the cuboid = n 

According to the question, 

Volume of Cuboid = Volume of n small cubes

$l\times b\times h=n\times a^3$

$10\times 6\times 4=n\times (2{)}^3$

$n=\frac{240}{8}=30$

Hence, the number of cubes that can be cut out from the given cuboid is 30.