Where is pi used today?

Answer 1

In a lot of different places!

Obviously in geometry. It occurs in the formulas for the circumference and area of a circle, and for the surface area and volume of spheres, cylinders and cones. In mechanics, the formulas for moments of intertia of spheres, cylinders, and so on, involve pi.

Pi also occurs in trigonometry, and is involved in the definition of radian measure (2 pi radians = 360 degrees). Radian measure is used when trigonometric functions are combined with calculus, and that combination has connections with many parts of physics and engineering.

For instance, Fourier series arise from the combination of trigonometry and calculus. Fourier used them to solve problems connected with the distribution of heat in physics. Now they are used in all sorts of places, including acoustics, image analysis, and engineering problems like analysing the behaviour of the digital-to-analogue converters used in CD players and portable audio devices. The formulas for Fourier series involve pi.

There is also a connection between trigonometry and complex numbers, shown in Euler's identity e^(i pi) = -1 (where ^ means "to the power of"). Again, complex numbers are used widely in physics and engineering.

Another place that pi occurs is in probability and statistics: the formula for the normal distribution, which is used all the time in probability and statistics, involves pi.

There is more ...

In summary, if you look at almost any technical field that has some mathematics at its base, you will find pi.