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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 is 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.
(If there is no image below, see support page.)

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.

Formula for 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:
Circle. One vertical line through the center, one horizontal across the upper part, each half labelled 'a' Vertical line labeled B in top part, c in bottom 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.

Derivation

This is actually a use of the intersecting chord theorem. In the figure on the right the two lines are chords of the circle, and the vertical one passes through the center, bisecting the other chord.

Recall from the intersecting chord theorem that Equation: a times a equals b times c Since a is half the arc's width W, and b is its height H: Equation: w over 2 times w over 2 equals H times c Or Equation: w squared over 4 equals h times c Dividing both sides by H Equation: c equals w squared over 4 H

Now, the diameter of the circle is equal to b+c, and b is the height H, so Equation: diameter = h + w squared over 4 h The radius is half the diameter: Equation: radius = h over 2, plus w squared over 8 H

To find the center of the arc, you simply measure down from the top of the arc by an amount equal to the radius, measuring down a line perpendicular to the chord defining the base of the arc.

By Construction

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. See Constructing a circle through three points.

Other circle topics

General

Angles in a circle

Arcs

(C) 2007 Copyright John Page