Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Circular arc
Summary
A circular arc is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than pi radians (180 degrees); and the other arc, the major arc, subtends an angle greater than pi radians. The arc of a circle is defined as the part or segment of the circumference of a circle. A straight line that connects the two ends of the arc is known as a chord of a circle. If the length of an arc is exactly half of the circle, it is known as a semicircular arc.
Arc length#Arcs of circles
The length (more precisely, arc length) of an arc of a circle with radius r and subtending an angle θ (measured in radians) 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 = α/180pi, 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 in degrees/360° = L/circumference.
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
Circular sector#Area
The area of the sector formed by an arc and the center of a circle (bounded by the arc and the two radii drawn to its endpoints) is
The area A has the same proportion to the circle area as the angle θ to a full circle:
We can cancel pi on both sides:
By multiplying both sides by r^2, 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
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 .
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.