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Notes

\(\pi\) is the ratio of the circumference of a circle to its diameter. It is an irrational number with approximate value

\( \pi \approx 3.1415926535897932384626433832795028841971… \)

Here are a few formulas for calculating the value of pi:

Leibniz formula: \(\pi = \frac{4}{1} – \frac{4}{3} + \frac{4}{5} – \frac{4}{7} + \frac{4}{9} – \frac{4}{11} + \dotsb\)

Madhava–Leibniz series: \(\pi = \frac{4}{1} – \frac{4}{3} + \frac{4}{5} – \frac{4}{7} + \frac{4}{9} – \frac{4}{11} + \dotsb\)

Wallis formula: \(\pi = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \dotsb\)

Euler’s formula: \(\pi = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \dotsb\)

Monte Carlo method for approximating pi: This method involves randomly generating points within a square and counting the number of points that fall within a quarter circle inscribed within the square. The ratio of the number of points within the quarter circle to the total number of points can be used to approximate the value of pi.

It’s worth noting that these formulas can be used to calculate the value of pi to various degrees of accuracy, depending on the number of terms used in the series.

See Related Pages\(\)

\(\bullet\text{ Geometry Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Area and Circumference of Circles}\)
\(\,\,\,\,\,\,\,\,\)\(\,\, A= \pi r^2, \,\,C=2 \pi r…\)

In Summary

Pi, denoted by the symbol π, is a mathematical constant approximately equal to 3.14159. It represents the ratio of the circumference of a circle to its diameter, which is a constant value for any given circle. In other words, if you measure the distance around a circle (the circumference) and divide it by the distance across the circle (the diameter), you will always get the same number, which is approximately equal to 3.14159.

Pi is an irrational number, which means that it cannot be expressed as a simple fraction and has an infinite number of decimal places. Despite this, it has been calculated to over one trillion decimal places and is a fundamental constant in mathematics. It appears in many different areas of mathematics and science, including geometry, trigonometry, and physics.

Pi has been known to humans for thousands of years, and it has been studied and celebrated by mathematicians and others throughout history. In fact, the ancient Greeks used the letter “π” to represent the number, and it is still used today. There is even an annual celebration called Pi Day on March 14 (3/14) to honor the number and its importance in mathematics. People often have pi-themed parties, hold pi recitation contests, and eat pie on this day.

The concept of pi is typically introduced in geometry classes, specifically in lessons on circles. In a geometry class, students may learn about the properties of circles and how to use pi to calculate the circumference and area of a circle.

However, pi is also used in other math subjects beyond geometry. For example, it appears in trigonometry, which is the study of triangles and the relationships between their sides and angles. In physics, pi appears in formulas related to circular motion, such as the period of a swinging pendulum.

While pi has an infinite number of decimal places, in most practical situations, it is sufficient to round it to a few decimal places. For example, in everyday calculations, it is common to use 3.14 for pi. However, in more precise calculations, such as in scientific or engineering applications, it may be necessary to use more decimal places.

Real world examples of The number π

Measuring circles: Pi is used to calculate the circumference and area of circles. For example, if you want to find the circumference of a wheel on a car, you would use the formula C = 2πr, where C is the circumference, π is the constant pi, and r is the radius of the wheel.

Trigonometry: Pi is used in trigonometry to calculate the lengths of sides and angles in triangles. For example, the sine, cosine, and tangent functions all involve pi.

Physics: Pi appears in many formulas in physics, particularly in formulas related to circular motion. For example, the formula for the period of a swinging pendulum (the time it takes for the pendulum to swing back and forth) involves pi.

Engineering: Pi is used in engineering to calculate the dimensions and properties of circular objects, such as gears, wheels, and pipes.

Computer Science: Pi is used in computer science to calculate the area and circumference of circles in computer graphics and simulations.

The concept of pi has been known to humans for thousands of years. The ancient Egyptians, for example, used a value close to pi in their calculations of the area of circles, and the ancient Greeks also had a good understanding of the number. Archimedes was one of the first mathematicians to give a precise value for pi and to prove that it is an irrational number (meaning that it cannot be expressed as a simple fraction). He did this by using a method called exhaustion, in which he calculated the perimeter of a regular polygon with more and more sides and showed that the value approached the circumference of a circle as the number of sides increased. Since Archimedes’ time, pi has been studied and calculated to greater and greater precision by mathematicians around the world. The modern symbol for pi (π) was first used by the Welsh mathematician William Jones in 1706. Jones used the symbol in a book called “Synopsis Palmariorum Matheseos,” which introduced the concept of pi as the ratio of the circumference of a circle to its diameter. Today, pi has been calculated to over one trillion decimal places and is a fundamental constant in mathematics.

Topics working with π

Geometry: Pi is a fundamental concept in geometry, which is the study of shapes and their properties. In geometry, students learn about circles and how to use pi to calculate the circumference and area of a circle.

Trigonometry: Pi is also used in trigonometry, which is the study of triangles and the relationships between their sides and angles. Trigonometry involves the use of functions such as sine, cosine, and tangent, which all involve pi.

Calculus: Calculus is a branch of mathematics that involves the study of rates of change and the accumulation of quantities. Pi appears in many calculus formulas, such as the formula for the area of a circle (A = πr^2) and the formula for the volume of a sphere (V = 4/3πr^3).

Physics: Pi appears in many formulas in physics, particularly in formulas related to circular motion. For example, the formula for the period of a swinging pendulum (the time it takes for the pendulum to swing back and forth) involves pi.

Engineering: Pi is used in engineering to calculate the dimensions and properties of circular objects, such as gears, wheels, and pipes.

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