Geometry - Volume
- What is Volume?
- Three-Dimensional Shapes and Their Volumes
- Different Ways of Volume Measurement
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What is Volume?
Volume is the amount of space occupied by an object or a substance in three-dimensional space and is typically expressed in cubic units such as cubic meters `(\text{m}^3)` or cubic centimeters `(\text{cm}^3)`.
Volume can also be the amount of space occupied by a liquid or a gas in a container.
There are many practical examples of volume around us such as the bucket filled with some amount of water, the amount of air present in a room, the volume of parcels in shipping containers, etc.
Three-Dimensional Shapes and Their Volumes
We have numerous three-dimensional shapes around us. For example, Rubik’s cube is a three-dimensional shape, a golf ball, a shuttle cock, etc.
The three-dimensional shapes are characterized by vertices, edges, and faces. The fundamental three-dimensional shapes are cube, rectangular prism, cylinder, sphere, cone, pyramid, and triangular prism. Other shapes also exist such as octahedron, dodecahedron, etc.
Cube
A cube is defined as a three-dimensional shape with six faces (congruent squares), `12` edges, and eight vertices. The most important properties of a cube are its symmetry and uniformity.
Below is the image of a cube.
A cube has all the sides equal. In the above figure, “`s`” denotes the length of one side of the cube. Therefore, all the sides of the cube are of measure “`s`”.
The volume of the cube is given by the formula.
Volume of cube `=s^3`
Practical examples of a cube are dice, sugar cubes, ice cubes, Rubik cube, etc.
Rectangular Prism
A rectangular prism is a three-dimensional figure with `6` faces, `12` edges, and `8` vertices. A rectangular prism usually has a length, breadth, and height of different measures. A rectangular prism with length ‘`l`', width ‘`w`’, and height ‘`h`’ is shown below.
The volume of a rectangular prism is calculated using the formula below.
Volume of rectangular prism `= l\times b\times h`
Practical examples of a rectangular prism are a shoe box, pizza box, laptop, books, etc.
Cylinder
A cylinder is a three-dimensional shape that has two flat circular faces connected by a curved surface.
A cylinder can be measured using the radius of the flat circular faces and the height of the curved surface between the flat circular faces. A cylinder with a radius ‘`r`’ and height ‘`h`’ is shown in the image below.
The volume of a cylinder is calculated by multiplying the area of the base with the height of the cylinder.
Volume of cylinder `= \pi r^2h`
Practical examples of cylinders are coke cans, pringles containers, fire extinguishers, etc.
Cone
A cone is a three-dimensional shape with a flat circular base connected to a curved surface converging at the vertex or apex above the flat circular face.
The apex is directly above the center of the circular face. A cone can be measured by the radius of the flat circular face and the height of the cone from the apex to the center of the flat circular surface. Also, any cone has one more dimension which is called slant height. The dimensions are shown in the image below.
- `h` is the height of the cone.
- `l` is the slant height of the cone
- `r` is the radius of the cone.
The volume of the cone is calculated using the below formula.
Volume of cone `=1/3 \pi r^2h`
Practical examples of cones are cone ice cream, birthday hats, volcanoes, etc.
Sphere
A sphere is a three-dimensional shape that is round in shape as well as symmetrical in all directions. A sphere has a radius, diameter, volume, and surface area as well. A sphere can be measured by its radius. All other measurements such as diameter, volume, and surface area are based on radius.
A sphere with a radius ‘`r`’ is shown in the below image.
The volume of the sphere is calculated using the below formula.
Volume of sphere `=4/3 \pi r^3`
Practical examples of the sphere are numerous such as the Sun, Moon, Earth, tennis balls, marbles, etc.
Different Units of Volume
There are various units of volume that we use based on the substance we are measuring the volume of. Some of the most commonly used units are cubic meter, cubic feet, liters, gallons, pints, quarts, cups, milliliters, etc. Let us learn some of the measurement units one by one.
- Liter: A liter is the basic unit of measurement, and it is equal to `1,000` cubic centimeters `(\text{cm}^3)`. It is denoted by ‘`l`’.
- Milliliter: A milliliter is a sub-unit of liter. It is given as `1/(1000)` of a liter. It is used for small measurements such as medicines. It is denoted by ‘`\text{ml}`’.
- Cubic centimeter: A cubic centimeter is equivalent to one milliliter. It is used for measurement in science experiments. It is denoted by `\text{cm}^3`.
- Cubic meter: If the dimensions of a three dimensional shape are measured in meters, the volume of the shape can be calculated in `\text{m}^3`.
- Gallon: A gallon is a unit of measurement for liquids that is commonly used to measure amount of milk, amount of gasoline etc. `1` gallon is approximately equal to `3.8` litres.
Summary
The following table shows the shapes and their formulae for volume.
Shape | Formula |
Cube | Volume of cube `=s^3` |
Rectangular Prism | Volume of rectangular prism`= l\times b\times h` |
Cylinder | Volume of cylinder `= \pi r^2h` |
Cone | Volume of cone `= 1/3 \pi r^2h` |
Sphere | Volume of sphere `=4/3 \pi r^3` |
Solved Examples
Example `1`: A trailer is filled with cubic containers. Each container is `3` m long. What is the volume of one such container?
Solution:
Here side length of the container is given as `s=3` m
Then, volume of the container`=s^3=3^3=3\times 3\times 3=27 \text{m}^3`
Example `2`: Emily wants to paint a wall and has a cylindrical bucket of radius `4` cm and a height of `6` cm. How much paint is the cylindrical bucket holding? Consider `\pi = 22/7`.
Solution:
The radius `(r)` of the bucket is `4` cm.
The height `(h)` of the bucket is `6` cm.
Volume of cylinder `= \pi r^2h=(22)/7\times 4\times 4\times 6=301.7\ \text{cm}^3`
So, the bucket is holding `301.7\ \text{cm}^3` of paint.
Example `3`: Organizing a birthday party is fun. Natalie bought a birthday party popper shaped like a cone. The radius of the popper’s base is `6` cm and the height is `21` cm. How much confetti does the party popper contain? Consider `\pi = 22/7`.
Solution:
The radius (\(r\)) of the popper is \(6\) cm.
The height (\(h\)) of the popper is \(21\) cm.
The volume of the popper cone \(= \frac{1}{3} \pi r^2 h\)
The volume of the popper cone \(= \frac{1}{3} \times \frac{22}{7} \times 6\times 6 \times 21 = 792 \, \text{cm}^3\)
So, the party popper contains \(792 \, \text{cm}^3\) of confetti.
Example `4`: A toddler’s toy is a spherical ball of radius `4` cm inside a cubic box of side `7` cm. Find the total volume of the ball and the cubic box. Consider `\pi = 22/7`.
Solution:
Given: \( r = 4 \) cm, \( s = 7 \) cm
Volume of sphere \( = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (4^3)= \frac{4}{3} \times \frac{22}{7}\times 4\times 4\times 4 = 268.19 \, \text{cm}^3 \)
Volume of cube \( = s^3 = 7^3 = 343 \, \text{cm}^3 \)
Total volume \( = \) Volume of sphere \( + \) Volume of cube
Total volume of the ball and the cubic box \( = 268.19 + 343 = 611.19 \, \text{cm}^3 \).
Practice Problems
Q`1`. A spherical tank of a radius of `3` meters can hold _____ l of water. Consider `\pi = 3.14`.
- `113040`
- `114000`
- `125000`
- `130000`
Answer: a
Q`2`. An ice cream cone of radius `5` cm and height `7` cm can contain ______ cubic centimeters of ice cream. Consider `\pi = 22/7`.
- `194.35`
- `200`
- `183.33`
- `130`
Answer: c
Q`3`. Erica has a cylindrical container full of slime. The radius of the container is `10` cm and the height is `12` cm. What is the amount of slime inside the container? Consider `\pi = 3.14`.
- `2334\ \text{cm}^3`
- `3768\ \text{cm}^3`
- `3218.811\ \text{cm}^3`
- `1200\ \text{cm}^3`
Answer: b
Q`4`. A mason is building a pool of rectangular shape with dimensions of a length of `10` meters and a breadth of `8` meters. The owner of the pool decided that the depth of the pool must be not more than `2` meters as they have a `3`-year-old child. What is the volume of water that the pool can hold?
- `200\ \text{m}^3`
- `260\ \text{m}^3`
- `390\ \text{m}^3`
- `160\ \text{m}^3`
Answer: d
Frequently Asked Questions
Q`1`. Are there any other three dimensional shapes except those mentioned in the article?
Answer: Yes, there are a lot of other three dimensional shapes. For example: triangular prism, pyramid, hexagonal prism, pentagonal prism, octahedron, torus, etc.
Q`2`. Are cube and rectangular prisms the same thing?
Answer: A cube is a special case of a rectangular prism where all its sides are of equal length.
Q`3`. Which units of volume measurements are most commonly used in the world?
Answer: Volume can be measured by different units such as `\text{cm}^3,\ \text{m}^3,` liters, and gallons, depending upon the type of application and the region where units are used. For example in the U.S. volume is expressed in gallons as well as in barrels.
Q`4`. Can I calculate the volume of a composite shape?
Answer: Yes, you can calculate the volume of a composite shape. For this, you have to calculate the volume of each shape individually and then add or subtract the volumes according to the problem.