Abhishek Verma 💻

What I love about mathematics is that a simple idea can be extended to unexpected limits and it may still work perfectly. Moreover, it often leads to further insights ! For instance, as number of sides in regular polygon are increased, it approaches circle. This simple idea can be used to calculate area of circle (from area of polygon) as well as derive value of π. Suppose the regular polygon is inscribed in a circle of radius "r". An n-shaped polygon can be split into n triangles (△). Base of △ = 2 r sin(π/n) Height of △ = r cos(π/n) Area of △ = (1/2) * base * height = (1/2) * 2 r sin(π/n) * r cos(π/n) = (1/2) r² sin(2π/n) ☑️ So, area of n-sided polygon = n * area of △ = (n/2) r² sin(2π/n) ✅ Now, if the number of sides 'n' is increased infinitely, that is, apply lim(n→∞), it results into area of circle = lim(n→∞) (n/2) r² sin(2π/n) = r² lim(n→∞) (n/2) sin(2π/n) = π r² ✅ With "r=1", this area would equate to π. In fact, this method (calculating area of polygon) had been used to improve value of π for several centuries!

👋 Need some suggestions. Few weeks back, I planned to write threads on the topics below (already posted 6 of them). If you could choose just 1 of the remaining 4(bottom) which one would you like to read ? Feel free to drop any other topic of your choice too. 🙂 1/2

Actually, infinitely many such equations can be framed. We can start with unknown values x, y & z. I'm not mentioning every step rigorously here, but it's enough to figure out. The final equation on left implies x=y and z=x²-1. Hence, we get values for all x>1