# Volume Formula

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### Volume Formula:

Volume is defined as the space occupied within a 3-dimensional solid object. Volume of an object is also referred to as its Capacity. Volume is one of the most important and commonly used measuring formula to calculate the space enclosed within the boundaries of a 3-dimensional object. The objects which have 3 dimensions of length, width and height are known as a 3-dimensional object. Examples of such figures include cube, cuboid, prism, sphere, cone, pyramid etc. The standard measuring units for volume are cubic centimeters, cubic meters, cubic foot, cubic inch, liters, etc. The figure below shows us various 3-dimensional objects and the spaces enclosed within them. Before we calculate the volume of each of the above figures, we must know how to convert one unit of volume to another unit.
Conversion of Volume:
In order to convert volume from one unit to another unit, we should be aware of the standard conversion between the units.
For example, if we want to convert 1 cubic yard to cubic feet, then we must first know the standard conversion between yard and feet.
1 yard = 3 feet
Now, a volume of 1 cubic yard implies 1yard * 1 yard * 1 yard
Now to convert it to cubic feet and considering 1 yard is equal to 3 feet, we get:
Volume of 1 cubic yard implies = 3 feet * 3 feet * 3 feet = 27 cubic feet.
Therefore, 1 cubic yard = 27 cubic feet.

Example: Convert 5 cubic meters to liters.
The standard unit conversion between cubic meters and liters is: 1 cubic meter = 1000 liters
Hence, 5 cubic meters = 5 * 1000 = 5000 liters.

### Volume Formula for Various Shapes:

Calculating volume is a very commonly done procedure, and this process is done in our daily life quite often. If we need to measure the space within a rectangular box, then we must know the volume formula for that shape and similarly to be able to know the amount of water that can fit into a bottle, we must measure the space enclosed within the bottle.
Now let us calculate the volume formula for different geometry shapes with different sizes.
1)Volume of a Box (Cuboid):
A box is also considered a Cuboid or a rectangular prism, and the volume of a box is the volume or space enclosed within a box. As shown in the figure below, a box figure has length, width and height at right angles and the faces of the box or cuboid are usually rectangles. A cuboid as shown in the figure below has 6 faces, 8 vertices and 12 edges. If the dimensions of the box are given as length l, width w and height h, then the Volume of the box is given by the formula as shown below:
Volume of the Box, V = length * width * height. Therefore, V = lwh
Example: Find the volume of the box whose length is 12 inches, width is 8 inches and height is 7 inches.
Given that length, l = 12 inches
Width, w = 8 inches
Height, h = 7 inches
Therefore, Volume of the Box, V = length * width * height ==> V = 12 * 8 * 7 = 672 inches3
This implies, Volume of the Box = 672 cubic inches.

2)Volume of a Cube:
A cube is a 3-dimensional figure whose faces are all square faces. A cube has 6 congruent square faces, 12 edges and 8 vertices. A cube is also known as a regular hexahedron. Since a cube consists of all equal or congruent squares, hence its side are of equal length as well.
If the side length of a cube is ‘s’, then the Volume of a Cube, V = s * s * s. Therefore, V = s3
Example: What is the volume of a cube of side length 5cm?
Given the side length of a cube, s = 5cm
Volume of a Cube, V = s3 ==> Therefore, Volume, V = (5cm)3 = 125cm3 or 125 cubic centimeters.

3)Volume of a Cone:
A cone is a geometric shape which has a circular base, and a single vertex (also known as ’apex’). If the apex of the cone is exactly over the center of the circular base, then such a cone is known as the Right Cone. If the apex of the cone is not exactly over the center of the base, then such a cone is known as the Oblique Cone. For a given Cone, if the radius of the circular base is ‘r’, and height (or altitude) is ‘h’, then the Volume of the Cone is given by the equation below:
Volume of a Cone, V =1/3 * π * (radius)2 * height ==> Volume, V = 1/3 * π * r2 * h
Example: What is the Volume of a Cone whose radius is 7 inches and height is 8 inches?
Given that radius, r = 7 inches and height, h = 8 inches.
Volume of the Cone, V = 1/3 * π * r2 * h = 1/3 * 3.14 * 72 * 8 = 410.3 inches3

4)Volume of a Sphere and a Hemisphere:
A sphere is a 3-dimensioanl geometric figure which has no edges (sides) or vertices (corners). It is a perfectly symmetrical shape which has all its points on the surface equidistant from the center. If a sphere has a radius ‘r’, then the Volume of the Sphere is given by the equation below:
Volume of a Sphere, V = 4/3 * π * (radius)3 ==> Volume, V = 4/3 * π * r3

When a sphere is cut into half, then we get a Hemisphere.
Volume of a Hemisphere = 1/2 * (Volume of Sphere) = 1/2 * 4/3 * π * r3
Therefore, Volume of a Hemisphere, V = 2/3 * ???? * r3
Example: Find the volume of a sphere whose radius is 10m. If the sphere is cut into half, then what is the volume of the hemisphere?
Given that the radius of the sphere, r = 10m
Volume of the Sphere, V = 4/3 * π * r3 = 4/3 * 3.14 * (10)3
Therefore, Volume of the Sphere, V = 4186.7 m3

If the sphere is cut into half, we get a hemisphere.
Volume of a Hemisphere = 2/3 * π * r3 = 2/3 * 3.14 * (10)3
Therefore, Volume of a Hemisphere, V = 2093.3 m3

5)Volume of a Cylinder:
A cylinder is a solid closed figure which has two congruent parallel circular bases that are connected by a curved surface. A cylinder has 2 bases that are circular in shape and are parallel to each other. Since they are circular, the bases have radius. These circular bases are connected by a curved surface and the line joining the centers of the circular bases is known as the ‘Axis’. The perpendicular distance between the 2 circular bases is known as the ‘height’ or the ‘altitude’. In our daily life, a cylinder is a very commonly observed geometric shape. We usually find cylinders which are called Right Cylinders and the ones which are slant in their position, known as the Oblique Cylinders as shown in the diagram below. If the 2 bases are exactly above one another and the axis is at right angles to the base, then that cylinder is known as the Right Cylinder. If the 2 bases are not exactly above one another and the axis is not at right angles to the base, then that cylinder is known as the Oblique Cylinder. For a given Right or Oblique Cylinder, if the radius of the circular base is ‘r’ and height (or altitude) which is the perpendicular distance between the 2 bases is ‘h’ then the Volume of the Cylinder is given by the equation below:
Volume of a Cylinder, V = ???? * (radius)2 * height ==> Volume, V = π * r2 * h
Example: Calculate the Volume of a Cylinder whose radius is 6cm and height is 9cm.
Given that radius, r = 6cm and height, h = 9cm.
Volume of the Cylinder, V = π * r2 * h = 3.14 * 62 * 9 = 1017.36 cm3

6)Volume of a Pyramid:
A pyramid is a 3-dimensional solid which has a base (which can be of any polygon), and has three or more triangular faces meeting at a single vertex (also known as the ‘apex’). Volume of a pyramid is the amount of space contained within the pyramid.
Based on the base polygon we have different kinds of pyramids. If the base of a pyramid is a square, then we get a Square Pyramid. If the base of the pyramid is a triangle, then we get a Triangular Pyramid.

Volume of any Pyramid can be calculated using the formula, V = 1/3 * (Base Area) * (Height)

Volume of a Square Pyramid:
If the side length of the square base is ‘a’ and the perpendicular distance from the vertex (apex) to the center of the base is ‘h’, then the Volume of the Square Pyramid can be calculated by the equation below:
Volume of a Pyramid, V = 1/3 * (Area of the base) * HeightSo, therefore, Volume of the Square Pyramid V = 1/3 * a2 * h Example:Find the volume of the square pyramid whose side is 4m and height is 6m.
Given that the side length, a = 4m
Height, h = 6m
Volume of the Square Pyramid, V = 1/3 * a2 * h = 1/3 * (4)2 * 6
Therefore, volume of the Pyramid, V = 32m3

Volume of the Triangular Pyramid:
If the base of a pyramid is a triangle, then that pyramid is known as a Triangular Pyramid. Usually the triangle at the base has equal sides, thus making it an equilateral triangular pyramid or a regular triangular pyramid. But if the sides of the base triangle are not equal, then we get an irregular base pyramid. Volume of a Triangular Pyramid = 1/3 * Base Area * Height

Given the base of a Triangular pyramid is an equilateral triangle, and the side length is ‘a’ and the perpendicular distance from the vertex (apex) to the center of the base is ‘h’.
Since the base is an equilateral triangle of side length ‘a’, hence its base area is √3/4 * a2
Therefore, Volume of an Equilateral (Regular) Triangular Pyramid = 1/3 * √3/4 * a2 * h

Example: Calculate the volume of a regular triangular pyramid whose triangle side length is 5m and height is 8m.
Since it is mentioned that it is a regular triangular pyramid, hence the base is an equilateral triangle.
Given that the side length, a = 5m and height, h = 8m
Volume of an Equilateral Triangular Pyramid = 1/3 * √3/4 * a2 * h = 1/3 * √3/4 * 52 * 8 = 28.9 m3
Hence, the volume is 28.9 m3.

7)      Volume of a Prism:
A prism is a 3-dimensional solid object with flat faces which has identical bases (or ends) which are parallel to each other. The prisms have the same cross-section all along their length. Since a prism is a polyhedron, it does not contain any curved sides. All its faces are flat, and all the edges are straight lines. Based on the cross-section of the prism, there are various types of them. For instance, if the cross-section along the length of a prism is a square, it is a Square Prism. If the cross-section along its length is a triangle, then it is a Triangular Prism.   Volume of a Prism, V = Base Area * Height of the prism.
The base area of the prism depends on the type of base the prism has.

Volume of a Square Prism: Given a prism which has a square base of side length ‘s’, and the height of the prism is ‘h’.
Volume of the Square Prism = Base Area * Height ==> s2 * h
Volume of a Square Prism, V = s2 * h

Example: Calculate the volume of a square prism whose base length is 4m and height is 5m.
Given that the base side length, s = 4m and height, h = 5m
Volume of a Square Prism, V = s2 * h ==> V = (4m)2 * 5m = 80m3
Therefore, the volume of the given square prism is 80 cubic meters.

Volume of a Triangular Prism:
Given a prism which has a triangular base of side length ‘s’ and the height of the base triangle is ‘d’. The height of the prism is given to be ‘h’. Volume of a Triangular Prism = Base Area * Height
(Base Area of a triangle = 1/2 * base * height = 1/2 * s * d)
Therefore, Volume of a Triangular Prism, V = 1/2 * s * d * h

Example: Calculate the volume of a triangular prism whose base triangle length is 5m, height of the triangle is 3m and height of the prism is 8m.
Given that the base side length, s = 5m, height of the triangular base, d = 3m and height of the prism, h = 8m.
Volume of a Triangular Prism, V = 1/2 * s * d * h = 1/2 * 5m * 3m * 8m = 60m3
Therefore the volume of the given triangular prism is 60 cubic meters.