Proof. Since Ω is tangentially graph-like, for any w$_{1}$ ∈ T$_{γ}$ ( s$_{1}$ ) , w$_{2}$ ∈ T$_{γ}$ ( s$_{2}$ ), the angle between w$_{1}$ and w$_{2}$ is at most π $_{2}$. Since this is true for all q ∈ P , there is a point q ′ = γ ( s $^{′}$) ∈ P and w ′ ∈ T$_{γ}$ ( s $^{′}$) such that the angle between w ′ and tangent vectors for any other point q ∈ P is at most π $_{4}$. This means that P is the graph of a Lipschitz function g of rank 1 in the orientation defined by w $^{′}$. This does not necessarily imply that D ( q $^{′}$, r ) ∩ ∂ Ω, D ( p$_{1}$, r ) ∩ ∂ Ω or D ( p$_{2}$, r ) ∩ ∂ Ω is the graph of a Lipschitz function; we explore a Lipschitz condition for the disks in section 4.3. Let x$_{1}$, x$_{2}$ ∈ [ − r, r ] with p$_{1}$ = ( x$_{1}$, g ( x$_{1}$ )), p$_{2}$ = ( x$_{2}$, g ( x$_{2}$ )). Then the arclength from p$_{1}$ to p$_{2}$ is given by Lemma 4.2.18. If γ is tangent-cone graph-like with radius r and 0 ≤ s$_{1}$ ≤ s$_{2}$ < L with d ( γ ( s$_{1}$ ) , γ ( s$_{2}$ )) = δ < r , then the image of [ s$_{1}$, s$_{2}$ ] together with the straight line from γ ( s$_{1}$ ) to γ ( s$_{2}$ ) enclose a region with O ( δ $^{2}$) area. Proof. By Lemma 4.2.17, we have that the image of [ s$_{1}$, s$_{2}$ ] under γ has arc length s$_{2}$ − s$_{1}$ ≤ √$_{2}$$_{δ}$$_{. Therefore, the region of interest has perimeter at most (}$√$_{2 + 1)}$$_{δ}$ so by the isoperimetric inequality has area at most ( √ 2+1) 2 4 π δ 2 from which the conclusion follows. If Ω is tangent-cone graph-like with radius r , it can sometimes be nice to know that there is an approximating polygon to Ω which is also tangent-cone graph-like. The following lemmas explore this idea. Lemma 4.3.1. If ∂ Ω is TCGL with radius r then for each ϵ ∈ (0 , r ) , then there exists 95 ∫ x$_{2}$ x$_{1}$ $^{√}$1 + g $^{′}$( x ) 2 dx ≤ ∫ x$_{2}$ x$_{1}$ √$_{2}$$_{dx}$ = √$_{2(}$$_{x}$$_{2}$ − x$_{1}$ ) ≤ √$_{2}$$_{d}$$_{(}$$_{p}$$_{1}$$_{, p}$$_{2}$$_{)}$$_{.}$ 4.3 TCGL polygonal approximations