In this paper, we introduce and study a new geometric constant \(D_\varepsilon (X)\) ( \(\varepsilon \in [0,1)\) ) to measure the difference between approximate isosceles orthogonality and approximate Birkhoff-James orthogonality in real normed linear spaces \((X,\Vert \cdot \Vert )\) . A characterization of inner product spaces is provided in terms of the constant \(D_\varepsilon (X)\) . From the inequality concerning approximate isosceles orthogonality, we obtain both lower and upper bounds for \(D_\varepsilon (X)\) . More precisely, we show that \(\begin{aligned} \varepsilon +2(\sqrt{2(1-\varepsilon )}-1)\le D_\varepsilon (X)\le 1 \end{aligned}\) for all \(\varepsilon \in (0,2(\sqrt{2}-1))\) . We further demonstrate that the derived lower bound for \(D_\varepsilon (X)\) is sharp.
