In geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of the circumference of a circle. If the arc is part of a great circle (or great ellipse), it is called a great arc.
Substituting in the circumference
and solving for arc length, L, in terms of yields
An angle of α degrees has a size in radians given by
and so the arc length equals
A practical way to determine the length of an arc in a circle is to plot two lines from the arc's endpoints to the center of the circle, measure the angle where the two lines meet the center, then solve for L by cross-multiplying the statement:
- measure of angle/360 = L/Circumference.
For example, if the measure of the angle is 60 degrees and the Circumference is 24", then
- 60/360 = L/24
- L = 4".
This is so because the circumference of a circle and the degrees of a circle, of which there are always 360, are directly proportionate.
The area between an arc and the center of a circle is:
The area A has the same proportion to the circle area as the angle θ to a full circle:
We can get rid of a π on both sides:
By multiplying both sides by r2, we get the final result:
Using the conversion described above, we find that the area of the sector for a central angle measured in degrees is:
Arc segment area
The area of the shape limited by the arc and a straight line between the two end points is:
Using the equality in the intersecting chords theorem (also known as power of a point or secant tangent theorem) it is possible to calculate the radius r of a circle given the height H and the width W of an arc using:
- Definition and properties of a circular arc With interactive animation
- A collection of pages defining arcs and their properties, with animated applets Arcs, arc central angle, arc peripheral angle, central angle theorem and others.
- Weisstein, Eric W., "Arc" from MathWorld.
- Radius of an arc or segment With interactive animation
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