Height of the arc - formula

Calculate the height of the arc if the length of the arc is 77 and chord length 40.
Does exist a formula to solve this?

Correct answer:

h =  26.55

Step-by-step explanation:

α₀= 1.57079632679 = 90°; r = 49.0197224723; h = 14.358
α₁= 2.72236110757 = 155°58'47″; r = 28.2842712475; h = 22.399
α₄₉₈= 3.70059415974 = 212°1'42″; r = 20.8074694701; h = 26.548
α₄₉₉= 3.70059415974 = 212°1'42″; r = 20.8074694701; h = 26.548

$$\begin{gathered} 40=2r\sin (\alpha \mathrm{/}2) \\ 77=\alpha r \\ 40=2\cdot (77\mathrm{/}\alpha )\cdot \sin (\alpha \mathrm{/}2) \\ \alpha =2\cdot (77\mathrm{/}40)\cdot \sin (\alpha \mathrm{/}2) \\ {\alpha }_{n+1}=2\cdot (77\mathrm{/}40)\cdot \sin ({\alpha }_n\mathrm{/}2) \\ {\alpha }_{n+1}=3.85\cdot \sin ({\alpha }_n\mathrm{/}2) \\ \alpha =3.70059415974\doteq 212^{\circ }{1}^{\mathrm{\prime }}42\mathrm{"} \\ r=77\mathrm{/}\alpha =20.8074694701 \\ h=r(1-\cos \alpha \mathrm{/}2)=26.55 \end{gathered}$$

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