In Euclidean geometry, an arc is a connectedsubset of a differentiable curve. Arcs of lines are called segments or rays, depending whether they are bounded or not. A common curved example is an arc of a circle, called a circular arc. In a sphere, an arc of a great circle is called a great arc. Every pair of distinct points on a circle determines two arcs. If the two points are not directly opposite each other, one of these arcs, the minor arc, will subtend an angle at the centre of the circle that is less than radians, and the other arc, the major arc, will subtend an angle greater than radians.
The length of an arc of a circle with radius r and subtending an angle θ with the circle center — i.e., the central angle — is This is because Substituting in the circumference and, with α being the same angle measured in degrees, since θ = , 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: For example, if the measure of the angle is 60 degrees and the circumference is 24 inches, then This is so because the circumference of a circle and the degrees of a circle, of which there are always 360, are directly proportional. The upper half of a circle can be parameterized as Then the arc length from to is
The area of the sector formed by 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 cancel on both sides: By multiplying both sides by r, 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 bounded by the arc and the straight line between its two end points is To get the area of the arc segment, we need to subtract the area of the triangle, determined by the circle's center and the two end points of the arc, from the area. See Circular segment for details.
Arc radius
Using the intersecting chords theorem it is possible to calculate the radius r of a circle given the height H and the width W of an arc: Consider the chord with the same endpoints as the arc. Its perpendicular bisector is another chord, which is a diameter of the circle. The length of the first chord is W, and it is divided by the bisector into two equal halves, each with length. The total length of the diameter is 2r, and it is divided into two parts by the first chord. The length of one part is the sagitta of the arc, H, and the other part is the remainder of the diameter, with length 2r − H. Applying the intersecting chords theorem to these two chords produces whence so
