What is Pi?

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 1 Introduction to Pi 2 What is Pi? 3 What is Pi used for? 4 How to calculate Pi? 5 Formula for Calculating Pi 6 Value of Pi in Decimal and Fraction 7 Important Notes on Pi 8 Solved Examples on Pi 9 Tips and Tricks 10 Practice Questions on What is Pi 11 Maths Olympiad Sample Papers 12 Frequently Asked Questions (FAQs)

Pi – Introduction

The research on "what is pi" dates back to 2000 BC.

Pi (pronounced as "pie") is a mathematical constant. It is an irrational number.

It is defined as the ratio of the circumference of the circle to its diameter.

The pi symbol is $$\pi$$. It is also called Archimedes' constant.

The Greek mathematician, Archimedes, created an algorithm to approximate the pi value.

The simulation below shows the circumference of a circle whose diameter is $$1 \: \text{cm}$$.

What is Pi?

We all know that pi value is constant. It is an irrational number, and is usually, approximated to $$3.14$$

Let us now do a small activity to know what is pi exactly.

Draw a circle of any radius.

Use a thread and measure the diameter.

Now, bend this thread to form a curve and place it along the circumference of the circle.

You can also see this in the simulation below. Click on start animation to know how to do the activity

We observe that the diameter spans across the circumference three times and a little part of the circumference is left uncovered by our thread which is approximately $$0.14 \:\text{units}$$.

Repeat this experiment with different diameters in the figure above and observe that there is always a shortfall of approximately $$0.14 \: \text{units}$$ irrespective of the size of the circle.

This indicates that the circumference is approximately $$3.14$$ times the length of the diameter.

The number $$3.14$$ is the mathematical constant called the pi value.

What is Pi Used for?

Let us understand where and what is pi used for? Pi is used to calculate the area and circumference of circles.

Any circular shape is dependent on the pi value. $$\pi$$ is found in many formulae in trigonometry to examine the relationship between the lengths and angles of triangles and in geometry where we study about shapes, sizes, relative positions, and the properties of space.

It is also used extensively in the field of architecture and robotics. pi$$(\pi$$) is considered very important and hence, March 14 is celebrated as Pi day.

March 14 was the chosen day because March is the 3rd month and we can denote the date as 3-14 which resembles the pi value.

How to Calculate Pi?

Let us do a small activity to see how to calculate pi.

By the end of this activity, we will know what is pi exactly.

Draw a circle of diameter $$1 \:\text{cm}$$.

Now take a thread and place it along the border of the circle (the circumference).

Now place the thread on the ruler and note the length.

Repeat the process with diameters of $$2 \:\text{cm}$$, $$3 \:\text{cm}$$, $$4 \:\text{cm}$$ and  $$5 \:\text{cm}$$.

Record your observations in the table.

Diameter Circumference   \begin{align}\frac{\text{Circumference}}{\textbf{Diameter}}\end{align}
$$1 \: \text{cm}$$   $$3.1\: \text{cm}$$ $\frac{3.1}{1} = 3.1$
$$2\: \text{cm}$$   $$6.2\: \text{cm}$$ $\frac{6.2}{2} = 3.1$
$$3\: \text{cm}$$   $$9.3\: \text{cm}$$ $\frac{9.3}{3} = 3.1$
$$4\: \text{cm}$$   $$12.4\: \text{cm}$$ $\frac{12.4}{4} = 3.1$
$$5\: \text{cm}$$   $$15.5\: \text{cm}$$ $\frac{15.5}{5} = 3.1$

We can observe that the ratio of circumference to diameter is always the same. This answers our question what is pi ($$\pi$$).

Formula for Calculating Pi

Now that we have understood what is pi and we've observed that the pi value is constant, let us see how to calculate it.

Pi is the ratio of the circumference of a circle to its diameter. The formula for calculating $$\pi$$ is:

 \begin{align} \pi = \frac{\text{Circumference}}{\text{Diameter}} \end{align}

Remember this line to recall the value of pi.

 $$\pi = 3.141592....$$

Value of Pi in Decimal and Fraction

Value of Pi in Decimal

The value of pi is an irrational number.

It is non-terminating and non-recurring.

Pi value approximate to $$10$$ decimal place is $$3.141592653$$

For ease of calculations, it is often approximated to $$3.14$$

Value of Pi in Fraction

In the fractional form, the closest number to pi is \begin{align}\frac{22}{7} \end{align}

\begin{align}\frac{22}{7}\end{align} is a rational number.

On dividing \begin{align}\frac{22}{7},\end{align} we get the quotient which is a recurring decimal number but is close to the value of pi.

But, \begin{align}\frac{355}{133}\end{align} is more close to $$\pi$$ than \begin{align} \frac{22}{7}.\end{align}

Divide and compare it with pi value.

Important Notes
1. $$\ \pi$$ is a mathematical constant which is the ratio of the circumference of a circle to its diameter. It is an irrational number often approximated to $$3.14159$$.
2. Basic circle formulae involving $$\pi$$
a) $$\text{ Circumference}\!=\!\pi\! \!\times\!\text{Diameter}$$
b) $$\text{ Area}= \pi \text{r}^2$$

Solved Examples

 Example 1

A pizza of diameter $$14 \: \text{cm}$$ is divided into $$6$$ equal parts. What is the area of each slice?

Solution:

\begin{align}&= \frac{\text{Diameter}}{2}\\ &= \frac{14}{2} \\ &= 7 \:\text{cm} \end{align}

Area of the pizza

\begin{align} &= \pi\: r^2 \\ &= \frac{22}{7} \times 7 \times 7 \\ &= 154\: \text{cm}^2 \end{align}

Thus, total area of the pizza is $$154\: \text{cm}^2$$.

Area of a slice of the pizza

\begin{align} &= \frac{\text{Area of the pizza}}{\text{Number of slices}}\\ &=\frac{154}{6} \\ &= 25.66 \:\text{cm}^2 \end{align}

 $$\therefore \;$$ Area of a slice of pizza $$= 25.66 \:\text{cm}^2$$
 Example 2

John had to replace a leaky pipe. He measured $$44 \:\text{mm}$$ around the outside of the pipe. What is the diameter of the pipe he should buy?

Solution:

Circumference of the given pipe is $$44 \:\text{mm}$$

\begin{align} &= 2 \pi\text{r}\\ 44 &= 2 \times\pi \times \text{r} \\ 44 &= 2 \times \frac{22}{7} \times {r} \\ \therefore \text{r} &= 7 \:\text{cm} \end{align}

Therefore, diameter of the pipe

\begin{align} &= 2 \times \text{r}\\ &= 2 \times 7 \\ &= 14 \:\text{cm}\\ \text{D} &= 14 \:\text{cm} \end{align}

 $$\therefore$$ Diameter of the pipe $$= 14 \text{ cm}$$
 Example 3

Betty measured the diameter and circumference of a hula loop accurately.

The diameter was $$113 \:\text{cm}$$ and the circumference was $$355 \:\text{cm}$$.

She used these measurements and calculated $$\pi$$. Is her value close to the real value of $$\pi$$?

Solution:

We know that $$\pi$$

\begin{align} &= \frac{\text{Circumference}}{\text{Diameter}}\\ &= \frac{355}{113} \\ &= 3.1415929 \end{align}

 \begin{align} \frac{355}{113} \end{align} is very close to the pi value
 Example 4

Ria goes jogging in a park daily.

The park is shaped semicircular as shown. The diameter of the park is $$70\: \text{m}$$.

If she goes around the park $$10$$ times. How much distance does she cover?

Solution:

Perimeter (circumference) of the circle

\begin{align} &= \pi \times \text{d} \\ &= \pi \times 70 \\ &= 70 \pi \end{align}

Perimeter (circumference) of the semi circle

\begin{align} &= \frac{\text{circumference}}{2} + \text{diameter} \\ &= \frac{70 \pi}{2} + 70 \\ &= 35 \pi +70 \\ &= 35 \times \frac{22}{7} + 70 \\ &= 110 +70\\ &= 180 \: \text{m} \end{align}

Ria goes around the park $$10$$ times.

Therefore, the total distance covered by her

\begin{align} &= 180 \times 10 \\ &= 1800 \text{ m} \\ &= 1.8 \text { km} \end{align}

 Ria covers a distance of $$1.8 \:\text{km}$$.

Tips and Tricks
1. "How I Wish I Could Calculate Pi" - rembember this line to recall the value of pi (How - 3, I - 1, Wish - 4 , I - 1 Could - 5, Calculate - 9 Pi - 2) $$\pi = 3.141592$$
2. Based on the problem, for ease of calculation, use $$\pi$$ value as $$\frac{22}{7}$$ or $$3.141592$$.

Practice Questions

Here are a few problems related to what is pi.

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

1. Why is pi so important?

Any circular shape is dependent on the pi value.

Pi($$\pi$$) is very important in mathematical calculations in the field of Geometry, Trigonometry etc.

Pi is so important that the World celebrates March 14 as Pi day.

2. What is the value of pi $$(\pi$$) in decimal?

The value of pi to 5 decimal places is 3.14159

It is an irrational number, and never-ending and non-repetitive.

3. What is the value of pi $$(\pi$$) in fractions?

For ease of calculations, we normally use the $$\pi$$ value in fraction as $$\frac{22}{7}$$, but $$\frac{355}{133}$$ is more close to $$\pi$$ than $$\frac{22}{7}$$

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