Let $C$ be a smooth projective curve of genus $g \geq 2$. By Faltings Theorem, we know that there are only finitely many rational points on $C$. We compute the rational points on a family of genus 3 hyperelliptic curves which are curves of the form $y^2 = f(x)$ where $f(x)$ is a polynomial of degree $2g+1$ or $2g+2$ via the method of Dem’janenko-Manin.