Radius of an arc or segment

Definition: The radius of an arc or segment is the radius of the circle of which it is a part.
A formula and calculator are provided below for the radius given the width and height of the arc.
Try this Drag one of the orange dots to change the height or width of the arc. The calculated area is shown.

A window where the top is part of a circular arc
Circular arcs turn up frequently in the real world, such as the top of the window shown on the right. When constructing them, we frequently know the width and height of the arc and need to know the radius. This allows us to lay out the arc using a large compass.

To calculate the radius

Given an arc or segment with known width and height: Segment of a circle.  A horizontal base line with an arc on the top.  Its height is H and width of the base W The formula for the radius is: where:
W  is the length of the chord defining the base of the arc
H  is the height measured at the midpoint of the arc's base.


See How the arc radius formula is derived.


Height clear
Width clear
Radius clear

Enter any two values and press 'Calculate'. The missing value will be calculated. For example, enter the width and height, then press "Calculate" to get the radius. It works for arcs that are up to a semicircle, so the height you enter must be less than half the width.

Finding the arc width and height

The width, height and radius of an arc are all inter-related. If you know any two of them you can find the third. For more on this see Sagitta (height) of an arc

Using a compass and straightedge

A circle through any three points can also be found by construction with a compass and straightedge. This also yields the location of the center point, and hence its radius. In the applet at the top of the page, the three orange dots could be used in this method. See Constructing a circle through three points.

Other circle topics


Equations of a circle

Angles in a circle