Pi Formula: Definition, Calculation

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Pi is originally a Greek letter that is represented by the sign ‘π’. It is the ratio of the circumference and the diameter of the circle. The value of pi or π can easily be evaluated if the value of circumference and the diameter of the circle are known. In geometry, the pi formula or π is decimally represented as 3.14159…. But the exact value is not known as the decimal value is endless. However, using the pi formula we can easily find the value of circumference or the diameter if either of them is known. Here, we will be learning more about pi formula, formula, calculation and discuss some important questions.

Table of Content

What is the Pi formula
Formula for pi
Calculation of pi value
Facts About pi Formula
Things to Remember
Sample Questions

Key Takeaways: Pi, circle, geometry, circumference, diameter, radius.

Also read: Isosceles Triangle Theorems

What is the pi Formula?

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Pi, in geometry, is a constant that is related to the circumference and the diameter of any given circle. It is denoted by the sign π and its decimal value is about 3.141 (approximately). If we express Pi as the ratio of the circumference and the diameter of the given circle, we get,

π = \(\frac{Circumference}{Diameter}\)

The Formula for pi (π)

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The formula that is used to denote pi (π) is

Circumference = π × diameter

So, π or pi formula = circumference/diameter

Being an irrational number pi cannot be represented as a fraction. But it is often represented as 22/7, which is a non-terminating fraction. It is used for the calculation of pi or its approximation. It is amusing that irrespective of the size of the circle, dividing the circumference and the diameter of the circle always gives the value of pi (π).

Read more: Circumference of a circle

Calculation of pi (π) Formula

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The value of the pi (π) formula is calculated by the formula discussed above. Using the formula, π = circumference/diameter, the value of π is calculated. Let there be a circle of diameter 2 units. Place a thread along the border of the circle which is the circumference of the circle. After that, place that thread on the ruler and note the value.

Repeat this with other units of diameter as well, for example, 3 units, 4 units, etc. It is noticed that the value remains 3.1415 throughout. Hence, the value of pi is constant which is 3.14159.

Read more: Decimal expansion

Facts About pi Formula

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The first attempt to study and calculate pi was done by Rhind Papyrus.
The symbol pi (π) was introduced by William Jones in 1706.
The initial 36 digits of pi are known as Ludolphine numbers.
Pi day is celebrated on 14th March every year.
Surprisingly, there are no zeroes in the first 36 numbers.

Things to Remember

Pi formula is denoted by π and its decimal value is 3.14159.
The pi formula can be expressed as the ratio of the circumference and the diameter of any given circle, i.e. circumference/diameter.
It was introduced by Sir William Jones in 1706.
It is an irrational number. However, it is represented by some fractions such as 22/7.
The value of the pi (π) formula is constant and it does not depend on the size of the circle.

Sample Questions

Ques. Explain the pi (π) formula. (2 marks)

Ans: Pi formula is a constant that is widely used in geometry. It is denoted by “π”. Its decimal value is 3.14159. The fractional value of pi is 22/7 which is a recurring digit. It was first introduced by William Jones in 1706. Pi is expressed as the ratio of the circumference and the diameter of any given circle.

Ques. If the perimeter of a circular pipe as measured is 97 inches, evaluate the diameter of the pipe using the pi formula. (3 marks)

Ans: Given: The perimeter of the pipe is 97 inches.

Pi (π) is 3.14.

The Pi formula is:

Pi (π) = circumference/diameter

3.14 = 97/diameter

Diameter = 97/(3.14)

Diameter = 30.89 or 31 Inches to be precise.

Ques. What is the importance of pi formulas? (3 marks)

Ans: Following are the reasons for the importance of pi:

In mathematics, pi comes into use for the calculation of circumference and area of circles.
In trigonometry, pi is used for the calculation of different angles and lengths.
In geometry, pi is used for the study of shapes, sizes, and space properties.
Pi formula is of utter importance and hence marks a celebration on 14th March every year.

Ques. Calculate the value of the pi (π) formula if the value of circumference is 9.4245 and diameter is 3. (3 marks)

Ans: As the Pi formula is expressed as the ratio of the circumference of any given circle and the diameter of the same circle. So, given

Pi (π) = circumference/diameter

Pi (π) = 9.4245/3

Pi (π) = 3.14159.

Hence, the value of pi (π) is 3.14159 which is a constant.

Ques. Is the value of pi (π) never-ending? If so, why? (2 marks)

Ans: Yes, the value of pi is never-ending. Being an irrational number, it is non-terminating and does not have a fixed value. But it is often represented as 3.14 for ease in the calculations. The number after the decimal point is infinite, hence the value of pi (π) is never-ending.

Ques. Discuss is the value of pi in degrees. (2 marks)

Ans: The value of pi is said to be 180°. A complete revolution holds a 360° value, which is the circumference of the circle divided by its radius. That is 2πr/r. This gives 360° = 2π, and π equals to 180°. Hence, the value of pi (π) is 180° in radians.

Ques. Consider a circle with a diameter of 12cm. Find its area. (3 marks)

Ans: Given,

The diameter of the circle is 12 cms.

So the radius equals d/2 or 6 cms.

As known, the area of a circle is πr².

Area = π × r × r

Area = 22/7 × 6 × 6

Area= 22/7 × 36

Area = 113.14 cm².

The area of the given circle is 113.14 cm².

Ques. Consider a circle with a radius is 12 cm and its circumference as 75.5 cm. Find the value of pi. (3 marks)

Ans: Given: Radius is 12 cm and the circumference is 75.5 cm.

Diameter = 2 × radius

Diameter = 2 × 12 cm

Diameter = 24 cm

Pi (π) = circumference / diameter

Pi (π) = 75.5 / 24

Pi (π) = 3.14.

Ques. Consider a man walking around a playground with 25 cm as its diameter. What is the perimeter of the playground? (3 marks)

Ans: Given: The diameter of the playground is 25 cm.

Pi (π) = 3.14

Pi (π) = circumference / diameter

Circumference = pi (π) × diameter

Circumference = 3.14 × 25 cm

Circumference = 78.5 cm.

Mathematics Related links:

Signum Function	CoPlanar Vectors	Geometric Mean
Negative of a Vector	Inverse Trigonometry	Rolle’s Theorem
Cosine rule	Value of log1 to log 10	Tan2x Formula
Relations and Functions	Integration	Parabola formula
Factorial formula	Inverse Matrix formula	Bayes theorem formula

Pi Formula: Definition, Calculation

What is the pi Formula?

The Formula for pi (π)

Calculation of pi (π) Formula

Facts About pi Formula

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

2.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

3.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

4.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

5.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

6.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

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Pi Formula: Definition, Calculation

What is the pi Formula?

The Formula for pi (π)

Calculation of pi (π) Formula

Facts About pi Formula

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

2.Find:Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

3. Find: The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

4.Evaluate:\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

5. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

6. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

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SUBSCRIBE TO OUR NEWS LETTER

1.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

2.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

3.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

4.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

5.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

6.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).