  We study the James constant $J(\mathbb{X})$, an important geometric quantity
associated with a normed space $ \mathbb{X} $, and explore its connection with
isosceles orthogonality $ \perp_I. $ The James constant is defined as
$J(\mathbb{X}) := \sup\{\min \{\|x+y\|, \|x-y\|\}: x, y \in \mathbb{X},~
\|x\|=\|y\|=1 \}.$ We prove that if $J(\mathbb{X})$ is attained for unit
vectors $x, y \in \mathbb{X},$ then $x\perp_I y.$ We also show that if
$\mathbb{X}$ is a two-dimensional polyhedral Banach space then $J(\mathbb{X})$
is always attained at an extreme point $z$ of the unit ball of $\mathbb{X},$ so
that $J(\mathbb{X}) = \|z+y\| = \|z-y\|,$ where $ \| y \| = 1 $ and $z\perp_I
y.$ This helps us to explicitly compute the James constant of a two-dimensional
polyhedral Banach space in an efficient way. We further study some related
problems with reference to several other geometric constants in a normed space.
