Arc radius formula derivation
This page describes how to derive the forumula for the
radius of an arc given the arc's width W, and height H.
Derivation
The arc radius equation is 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.
The blue segment is the arc whose radius we are finding. Its width is 2a, and height b.
Recall from the intersecting chord theorem that
Since a is half the arc's width W, and b is its height H:
Combining terms
Dividing both sides by H
Now, the
diameter
of the circle is equal to b+c, and b is the height H, so
The radius
is half the diameter, so dividing through by 2:
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Other circle topics
General
Equations of a circle
Angles in a circle
Arcs
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