Let \(R^n\) be an \(n\)-dimensional Euclidean space and \((R^n,L)\) a Lagrange space with \(L\) a smooth function in the tangent bundle of \(R^n\) satisfying a certain regularity condition. For \(L=F^2\) with \(F\) homogeneous of degree one we have a Finsler space. Each tangent space \(R^n_x\) is also regarded as an \(n\)-dimensional Euclidean space with the rectangular coordinate system \(y=(y_1,\ldots,y_n)\). Then \(I_x=\{y\mid L(x,y)-1=0\}\) is a hypersurface in \(R^n_x\) called the indicatrix in \(x\). First, the authors give a self-contained proof of a formula which gives the Gaussian curvature of a hypersurface in \(R^n\) defined by a smooth function \(f\) in \(R^n\) in terms of \(f\) itself. This formula is applied to \(I_x\). Given a hypersurface \(S_x\) in each tangent space \(R^n_x\) a priori, there exists a Finsler space whose indicatrix is \(S_x\). Thus the Gaussian curvature of \(S_x\) is expressed in terms of Finsler geometry. As an example, a Randers space is treated.