# Dimensions

## What are Dimensions of a Shape

The dimension of an object/shape is basically a topological measure of the length, width, or height of its covering properties. Dimension is a measure of size extended in a specific direction. In simple terms, it is the number of coordinates required to specify a point on the object. For instance, a rectangle is a 2-dimensional figure, while a cube is 3-dimensional. The letter D in 2D, 3D denotes dimensions. The dimension of an object is sometimes also called its "dimensionality".

### Types of Dimensions

Most commonly, there are 4 types of dimensions as given below:

1. Zero Dimension: A point consists of no extent in any direction, so it has no dimension.

2. One Dimension: A line has how many dimensions? A line has only one dimension, which is its length.

3. Two-Dimension: A piece of paper consists of two dimensions i.e. length and width.

4. Three-Dimension: Dices consist of three dimensions i.e., length, width, and height. Dimension implies the measurement of the size or distance of an object or space in only one direction.

### Formula of Dimension

How many dimensions does a point have? Dimension is a measure of a line or point stretched out in one direction and we are aware that every shape around us consists of some dimensions. The theory of dimension does not contain any particular dimension formula. The value and the number of dimensions differ depending upon their size and shape. The following table depicts the dimension formulas of circles and polygons.

## Dimension Formulas

 Number of Dimensions Physical Quantity Dimension Formula Dimensions Used 2D - Polygons Area length × breadth (or) height length, breadth (or) height Perimeter Adding up all the distance around the object or shape distance 2D - Circle Area π r2 (or) π d radius (or) diameter Circumference 2 π r radius

Dimensions can be used to identify the unknown values of shapes using the help of formulas. For instance, if the perimeter of a square is 12 inches, we can determine its length with the help of the dimension formula: Perimeter = 6 × Length. Substituting the value of perimeter in the formula, we obtain 12 = 6 × Length. Thus, Length = 2 inches.

Different units are used to measure different quantities like length, tempurature, area etc. The following table shows the dimensions that are usually used for units like length, area, temperature, volume and weight.

## Dimensions of Units

 Quantity Units Length inches, feet, yards, miles Area Square Inch, Square Feet, Square Yard, Square Mile, Acre Temperature Celsius, Fahrenheit, Kelvin Volume (Capacity) quarts (qt), pint (pt), gallons (gal), fluid ounces (fl oz)

### How Many Dimensions Does a Plane Have

Planes are 2 dimensional. However, they can exist in 3-dimensional spaces as well. The plane contains two dimensions: length and width. But since the plane is limitlessly large, the length and width are unable to be measured. Just as a line is defined as a 1-dimensional figure or one-dimensional shape having two points, a plane is defined by three points.

### Dimensional Analysis

Dimensional analysis means to determine the physical quantities and evaluate the association between them. It helps us to study the nature of different objects like straightness and flatness mathematically.

### Dimensionless Quantities

The quantities that contain no dimension are known as dimensionless quantities. Such quantities consist of a certain value. For instance, the constant Pi - π is a dimensionless quantity since it has a fixed value. Following are the types of Dimensionless quantities:

1. Dimensionless Quantities With Units: These include angular displacement –Joule’s constant – joule/calorie, radian etc.

2. Dimensionless Quantities Without Units: These include pure numbers, π, sin θ, cos θ, tan θ, e, etc.

### Dimensional Formula

The expression displaying the powers to which the fundamental units are to be raised in order to get one unit of a derived quantity is known as the dimensional formula of that quantity.

If Q is the unit of a derived quantity denoted by the letter Q = MaLbTc, then MaLbTc is known as the dimensional formula and the exponents a, b and, c, are what we call the dimensions.

### Fun Facts

• It is believed that the Universe consists of 10 dimensions; however, fortunately, the human eye can only apprehend three.

• The meaning of dimension differs in mathematics and physics.

• Dimension is the point or value, measuring the object.

Q1. What are 3D and 2D Shapes?

Answer: 2-dimensional shapes are objects that consist of only two dimensions. For example, shapes like squares and rectangles are classified as 2-dimensional shapes. Whereas, shapes that consist of three dimensions are called 3-dimensional shapes. Cubes and cones are examples of 3-dimensional shapes.

Q2. Are There any Limitations of Dimensional Analysis?

Answer: Yes! Following are those limitations:

• Dimensionless quantities cannot be identified using this technique. Constant proportionality cannot be identified by this method. They can be identified either by theory or by experiment.

• This technique does not apply to trigonometric, logarithmic and exponential functions.

• With respect to physical quantities that are dependent upon more than 3 physical quantities, this method will be complicated.

• In some cases, the constant of proportionality also consists of dimensions. In such cases, we are unable to use this system.

• If one side of the equation consists of the addition or subtraction of physical quantities, we are unable to use this method in order to derive the expression.

Q3. What do we Understand by Dimensionless Variables?

Answer: Dimensionless variables are those physical quantities that neither contain any dimensions and nor do they have a fixed value. Examples of dimensionless variables include refractive index, the coefficient of friction, specific gravity, and Poisson’s ratio etc.