Arc Length Calculation Given Only the Chord Length and Arc Height (Sagitta)

Arc Length Calculation

How to calculate the length of an arc while only given the chord length and the sagitta or height of the arc at the midpoint perpendicular to the chord .
How to calculate the length of an arc while only given the chord length and the sagitta or height of the arc at the midpoint perpendicular to the chord . | Source

Important Note:

This calculation is only valid when the angle of the arc is less than or equal to 180 degrees. To calculate larger arc lengths, calculate the length of the smaller arc first, then subtract it from the circumference of the circle. This will give you the arc length of the remainder of the circle.

What application do you have for this calculation?

I for one needed to calculate the heat loss on a greenhouse design I was evaluating. I was deciding between a peaked roof style and a quonset (round roof) style.

I didn't know the radius of the roof curvature or the angle the arc spanned. I did however know the chord length which was the width of the greenhouse and the peak height of the greenhouse. To get the height of the arc, I subtracted the wall height from the peak height, Using the Pythagorean theorem, I was able to solve for the radius of the arc. Then using the law of signs I was able to solve for the angle of the arc.

Since the ratio of the arc length to the circumference of the circle is equal to the ratio of the arc angle to the full angle of the circle (360), I was able to plug the equations I had found for radius and arc angle, into the ratio equation and solve for L.

Now that I was able to calculate the arc length of the roof, I could multiply it by the length of the greenhouse to get the area of the roof that I need for my heat loss calculations

Please post your applications in the comments below.

Thank you..

Arc Length Calculation Derived

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Comments 6 comments

joanwz profile image

joanwz 3 years ago from Katy, Texas

Great explanation. The visuals helped a lot.


getting there profile image

getting there 3 years ago Author

Thanks for the feedback joanwz. I'm glad you found it useful.


paxwill profile image

paxwill 2 years ago from France

Very useful reference for constructing curved surfaces with given height and width.


getting there profile image

getting there 2 years ago Author

Thanks for the feedback paxwill. What sort of curved surfaces are you constructing?


mashiq546@yahoo.com 11 months ago

please i have 125 m curve length and 105 m chord length how to calculate do you have any formula for this question.


getting there profile image

getting there 11 months ago Author

What dimension are you trying to calculate?

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