# 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?

Result

h =  26.55

#### Solution:

α0= 1.5707963267949 = 90°; r = 49.019722472304; h = 14.358
α1= 2.7223611075682 = 155°58'47″; r = 28.284271247462; h = 22.399
α498= 3.7005941597357 = 212°1'42″; r = 20.807469470119; h = 26.548
α499= 3.7005941597357 = 212°1'42″; r = 20.807469470119; h = 26.548

$40 = 2 r \sin( \alpha/2) \ \\ 77 = \alpha r \ \\ 40 = 2 \cdot (77 / \alpha) \cdot \sin( \alpha/2) \ \\ \alpha = 2 \cdot (77 / 40) \cdot \sin( \alpha/2) \ \\ \alpha_{n+1} = 2 \cdot (77 / 40) \cdot \sin( \alpha_{n}/2) \ \\ \alpha_{n+1} = 3.85 \cdot \sin( \alpha_{n}/2) \ \\ \alpha = 3.7005941597357 \doteq 212^\circ 1'42" \ \\ r = 77/\alpha = 20.807469470119 \ \\ \ \\ h = r(1- \cos \alpha/2) = 26.55$

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