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Sagitta - The height of an arc or segment

The sagitta is the height of an arc.
It is the perpendicular from the midpoint of the arc's chord to the arc itself.
Try this Drag the orange dot to move the chord. Note how the sagitta is recalculated.

In the figure above, the blue arc is a portion of the circle that is cut off by the horizontal chord. The sagitta is the vertical line from the midpoint of the chord to the arc itself. It is a measure of the 'height' of the arc.

The length of the chord, sagitta and radius of the arc are inter-related, and if you know any two you can calculate the third.

Sagitta showing the radius of the arc and the chord length.  Depth of an arc

Finding the sagitta length S

You can find the length of the sagitta using the formula: where:

s is the length of the sagitta
r is the radius of the arc
l is one half of distance across the base of the arc (half the chord length)

Notice that there are two results due to the "plus or minus" in the formula. The smaller one is the sagitta as show in the diagram above. The other is the longer sagitta that goes the other way across the larger part of the circle:

Radius of the arc R

If you know the sagitta length and arc width (length of the chord) you can find the radius from the formula: where:
s is the length of the sagitta
r is the radius of the arc
l is one half of distance across the base of the arc (half the chord length)

Width of the arc (chord length) L

If you know the arc sagitta and radius you can find the arc's width, (which is the length of the chord) from the formula: where:
s is the length of the sagitta
r is the radius of the arc
l is one half of distance across the base of the arc (half the chord length)

Note In all the above formulae, the length l is half the width of the arc. The full width will be double this.

Calculator

Enter any two variables and leave one blank. When you press 'calculate' the missing one will be calculated. Remember to use the same units for all inputs.
ENTER ANY TWO VALUES
Sagitta S clear
Half chord length L clear
Radius R clear
   
 

Things to try

  • In the above applet click 'reset', and 'hide details'.
  • Drag the orange dot to reposition the chord.
  • Calculate the length of the sagitta.
  • Click 'show details' to check your result.

Other circle topics

General

Equations of a circle

Angles in a circle

Arcs