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".
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Types of Dimensions
Most commonly, there are 4 types of dimensions as given below:
Zero Dimension: A point consists of no extent in any direction, so it has no dimension.
One Dimension: A line has how many dimensions? A line has only one dimension, which is its length.
Two-Dimension: A piece of paper consists of two dimensions i.e. length and width.
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
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
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:
Dimensionless Quantities With Units: These include angular displacement –Joule’s constant – joule/calorie, radian etc.
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.
FAQs on Dimensions
1. What is the principle of homogeneity, and why is it important for dimensional analysis?
The principle of homogeneity states that an equation is dimensionally correct only if the dimensions of all the terms on both sides of the equation are the same. This principle is the foundation of dimensional analysis because it allows us to check the correctness of a physical equation, as we can only add or subtract quantities that have the same physical dimensions.
2. What are the seven fundamental quantities in physics and their dimensional symbols?
For your exams, it's crucial to know the seven fundamental quantities and their dimensional symbols as per the SI system. They are:
- Length (L)
- Mass (M)
- Time (T)
- Electric Current (A)
- Thermodynamic Temperature (K)
- Amount of Substance (mol)
- Luminous Intensity (cd)
3. How can I use dimensional analysis to check if a physics formula is correct?
To check a formula, you need to apply the principle of homogeneity. First, write down the dimensional formula for the quantity on the left-hand side (LHS) of the equation. Then, write the dimensional formulas for all the quantities on the right-hand side (RHS). Simplify the dimensions on the RHS. If the final dimensions on the RHS match the dimensions on the LHS, the formula is dimensionally correct.
4. What types of questions on dimensional analysis are frequently asked in Class 11 exams?
For the CBSE 2025-26 session, you can expect questions that test three main skills: 1) Checking the dimensional consistency of equations, 2) Deriving the relationship between different physical quantities, and 3) Converting a physical quantity from one system of units to another. You may also be asked 1-mark questions on the dimensions of physical constants or the limitations of this method.
5. If an equation is dimensionally correct, does it guarantee that the equation is physically correct? Why or why not?
No, a dimensionally correct equation is not necessarily a physically correct one. Dimensional analysis cannot determine the value of dimensionless constants (like 1/2, π, or 2). For example, the formulas for kinetic energy, K = ½ mv² and K = mv², are both dimensionally correct, but only the first one is physically right. Therefore, dimensional correctness is a necessary but not sufficient condition for an equation to be physically correct.
6. What are the key limitations of dimensional analysis that I should know for my exam?
Dimensional analysis has several important limitations that are often asked in exams. Key points to remember are:
- It cannot determine the value of dimensionless constants.
- It cannot be used for equations involving trigonometric, logarithmic, or exponential functions.
- It fails if a physical quantity depends on more than three other fundamental quantities.
- It cannot be used to derive a formula that contains the sum or difference of two or more quantities.
7. Besides checking formulas, what is another major application of dimensional analysis that is important for 3-mark or 5-mark questions?
A very important application is to derive the relationship between different physical quantities. If you know that a certain quantity depends on several other factors, you can use their dimensions to find the exact mathematical expression that connects them. For example, you can derive the formula for the time period of a simple pendulum by knowing it depends on its length (l) and acceleration due to gravity (g).
