You're likely familiar with the term 'pi' - it's so important in the mathematical world that it even has its own dedicated day in the United States (14th March)! However, you might be unsure about what it can be used for or how it is calculated.
There are many benefits to learning how to apply pi to different formulas so that you can use it to measure different structures and shapes. Pi isn't just used by mathematicians - it's used in most industries to some degree.
In this guide, we'll look at what pi is commonly used for, which mathematicians are credited with discovering, and why pi is considered an 'irrational number'.
Pi (π) is the ratio between the circumference and diameter of a circle. The ratio is a mathematical constant, meaning that the pi's value will stay the same, no matter how big or small a circle is.A circle's circumference divided by its diameter will always be approximately equal to 3.14159 . . . (pi).
The value of pi is an irrational number, which means that it cannot be expressed as the ratio between two integers. The digits of pi never repeat, and it's impossible to list the exact value because the numbers are never-ending.
3.14 or 22/7 are typically used to describe the approximate value of pi in calculations. British mathematician William Jones started using the symbol π in 1706 to use in place of the digits, and was made popular through the use of Swiss mathematician Leonhard Euler. The symbol is the 16th letter in the Greek alphabet and is pronounced as 'p' in both Ancient and Modern Greek.
The most common use for pi is to find a circle's area, circumference and radius. This is taught to most children while they are at school, as it's important to have a basic understanding of this area of mathematics. As circles have no straight edges, they cannot be easily measured and divided with rulers like other shapes can.
Whilst pi is often used in basic mathematics to measure the size of a two-dimensional circle, it is also commonly used by engineers and physicists for a variety of reasons, such as the following:
- Calculating the density of planets
- Measuring light and sound waves
- Constructing curved fixtures, such as pillars and pipes
- Designing pendulums for clocks
- Tracking and analysing population dynamics
Pi can be used to calculate the size of circular objects such as clock pendulums and pillars. However, pi can be applied to an object of any size, which means that it can also be used to calculate the size of planets. The density of any object (including planets) can be calculated using the formula Density (p) equals Mass divided by Volume (p=M/V).
Architects rely on pi when they are calculating the design and layout of buildings and structures. It can help them determine the size of a semi-circle under a bridge or arch and the size of a pillar or pipe. Knowing the size of these structures can help architects determine how much material is required for the construction and how much pressure will be put on each component of a building or structure.
The radius of a circle can be found by using the formula R = c ÷ (2 x π). This means that you must multiply pi by two and divide the circle's circumference by the answer.
The circumference of a circle is 34mm.
34 ÷ (2 x 3.14) = 5.41mm
If you don't know the circumference of a circle, but you know the diameter, you can calculate the radius by dividing the diameter by two.
To find the circumference when you know the diameter of a circle, you would need to multiply pi by the diameter (C = πd). Most calculators have the pi (π) button, or you can manually type in 3.14 to the equation.
The diameter of a circle is 24cm. In this instance, the equation is:
C = 3.14 x 24
C = 75.36
When you know the radius of the circle, you can work out the circumference by using the equation 2πr. This means you need to multiply two by pi and multiply that by the length of the circle's radius, as follows:
C = 2 x 3.14 x 12
C = 75.36
To find the area of a circle, you need to know the radius or diameter so that you can calculate A=πr2. This means that you need to multiply pi by the radius squared.
The radius of a circle is 10m.
A = 3.14 x 10 x 10 = 314m
A = 314m
If the diameter of a circle is 18cm, the radius would be 9cm, as the radius is half the length of the diameter.
A = 3.14 x 9 x 9 = 254.34cm
A = 254.34cm
Greek mathematician Archimedes of Syracuse is often credited for having 'invented pi' by doing the first approximate calculation of it in around 250 BC. Archimedes drew two polygons in and outside a circle and used the Pythagorean Theorem to work out the areas of the polygons. The area of the polygons gave the upper and lower bounds of the circle, which gave Archimedes an appropriation of pi. Although he didn't know the exact figure, he could deduct that pi had a value somewhere between 3 1/7 and 3 10/71.
A few hundred years later, Chinese mathematician Zu Chongzhi was able to calculate the ratio value between the circumference and the diameter of a circle to be 355/113.
It wasn't until the 1700s that the Greek letter 'π' began to be substituted for the digits of pi. Before the British mathematician William Jones used the symbol, pi was often approximated to 22/7 and 355/113, which led many people to believe that pi was a rational number. However, Jones theorised that pi was an infinite number with a non-repeating sequence that couldn't properly be expressed in a numerical form.
In the previous century, π was used by British mathematician William Oughtred to represent the circumference of a circle. This meant that π could vary depending on a circle's diameter, rather than stay constant as it does in today's form. Jones took Oughtred's writing and further developed the symbol so that it represented the irrational form of pi that we still use today.
As pi is an irrational decimal number, it has infinite digits. After the decimal point, the digits continue on forever. When calculating simple equations, pi is often simplified to its first three digits (3.14).
As technology evolves, it's becoming increasingly faster to calculate more digits of pi. In 2022, pi was calculated to 100 trillion digits by Google Cloud. The previous year, pi had been calculated to 62.8 trillion digits by scientists at the University of Applied Sciences of the Grisons, while 31.6 trillion digits were discovered in 2019 by Google Cloud.
Radians are one way of measuring angles and means that angles can be written down in simple formulas. Every circle has a ratio where the arc length ℓ of a circle is equal to the ratio between the angle θ that is subtended by the arc from the circle's centre and the angle in a single revolution.
This can be expressed in the following formulas (using π). However, the angles must be measured in radians rather than degrees:
ℓ/ 2πr = θ/2π
ℓ = rθ
The circumference of a circle is equal to 2πr, where r is the radius of a circle. This means that 360º (a full circle) is equal to 2π radians and 180º is equal to PI radians. A single radian is therefore 180/π degrees and one degree = π/180 radians.
To convert from degrees to radians, you would need to multiply the number of degrees by π/180 radians. For example, to covert 90º (a right angle) to radians, you would need to do the following:
90º = 90 × π/180 radians = π/2
To convert from radians to degrees, you need to multiply the number of radians by 180/π
Pi is an irrational number, which means that it cannot be expressed as the ratio of two integers. Most basic mathematical calculations simplify pi to its first three digits: 3.14 as the digits after the decimal point are infinite.
There are many uses for pi, from helping astronomers measure the density of planets to letting architects know how to construct a curved bridge. However, the most basic use for pi is to measure a circle's area, circumference and diameter. You can also use pi to measure angles when they are measured in radians.