# When Life Gives You Circles: Calculating the Square Root of Two

Poster - Applied

## Description

In this poster, we derive a method of approximating the square root of two. We do this by constructing a geometric figure that has inscribed circles lying across the diagonal of a square. By adding the radii of these circles, we can find the length of the diagonal, which will be the square root of two if the square had side length one. This project uses trigonometry, recursion, the binomial theorem, and a lot of algebra to arrive at the following equation: We find that the formula gives an approximation that is about 3 decimal places more precise for every iteration of this sequence. While other methods exist for calculating square roots more efficiently, geometric methods such as these are scarcely found and can apply to other problems. Specifically, these methods have potential applications in finding square roots generally, approximating pi, and describing other irrational numbers. Although space filling techniques have been used to find values, this method may be better as it uses circles. Additionally, it should be noted that this function has some properties which may lead to fast calculations using a program with optimizations. Namely, the 8s can be solved using bit manipulation and constructing Pascal’s triangle through recursion can eliminate the use of factorials.

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Apr 18th, 10:00 AM

When Life Gives You Circles: Calculating the Square Root of Two

Poster - Applied

In this poster, we derive a method of approximating the square root of two. We do this by constructing a geometric figure that has inscribed circles lying across the diagonal of a square. By adding the radii of these circles, we can find the length of the diagonal, which will be the square root of two if the square had side length one. This project uses trigonometry, recursion, the binomial theorem, and a lot of algebra to arrive at the following equation: We find that the formula gives an approximation that is about 3 decimal places more precise for every iteration of this sequence. While other methods exist for calculating square roots more efficiently, geometric methods such as these are scarcely found and can apply to other problems. Specifically, these methods have potential applications in finding square roots generally, approximating pi, and describing other irrational numbers. Although space filling techniques have been used to find values, this method may be better as it uses circles. Additionally, it should be noted that this function has some properties which may lead to fast calculations using a program with optimizations. Namely, the 8s can be solved using bit manipulation and constructing Pascal’s triangle through recursion can eliminate the use of factorials.