Here’s a fun formula for Pi involving an infinite product, known as Wallis’ Formula:

(Pi/2) = (2*2)(4*4)(6*6)/(1*3)(3*5)(5*7)

It is somewhat surprising that when you pull out every other pair of terms, you get a completely different kind of number!

Sqrt[2] = (2*2)(6*6)(10*10)/(1*3)(5*7)(9*11)

**Presentation Suggestions:**

You can also start with the infinite product, and ask if student can guess what it converges to, before you tell them the answer.

**The Math Behind the Fact:**

There is an infinite product formula for the sine function which yields Wallis’ formula as a consequence. Infinite products are defined as the limit of the partial products, which are finite. This is similar to the way we define infinite sums!

**How to Cite this Page:**

Su, Francis E., et al. “Wallis’ Formula.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**References:**

S. Vandervelde, *Mathematics and Informatics Quarterly*, 9 (1999), pp. 64-69.

**Fun Fact suggested by: **

Sam Vandervelde