Arc of a Circle

A circle arc is a section of the circumference defined by two points, A and B.
an example of an arc
These defining points, A and B, are referred to as the endpoints of the arc.

Simply put, an arc measures the curved distance between two distinct points on the circumference.

This differs from a segment, which measures the straight-line distance between the points.

Note: When two distinct points, A and B, on the circumference are selected, the circle is divided into two arcs that complete each other. Often, it's necessary to introduce a third point, C, to specify which arc is being referred to. By default, with just A and B mentioned, it's understood to be the arc that spans counterclockwise between them.

Calculating Arc Length

The length (L) of a circle arc is calculated by multiplying the central angle α (in radians) by the radius (r) of the circle. $$ L = r \cdot \alpha_{rad} $$ If the angle is in degrees, the arc length is determined by the formula: $$ L = r \cdot \frac{ \pi \cdot \alpha_{degrees} }{180} $$

Angles in degrees need to be converted to radians before applying the formula.

$$ \alpha_{rad} = \frac{ \pi \cdot \alpha_{degrees} }{180} $$

Then, substitute αrad into the formula for arc length.

$$ L = r \cdot \alpha_{rad} $$

Hence, the formula for arc length is:

$$ L = r \cdot \frac{ \pi \cdot \alpha_{degrees} }{180} $$

An arc is termed a full arc when it encompasses the entire circumference, with a length of 2πr, where r is the radius. $$ L = 2 \pi r $$

Note: This formula reflects the proportionality between the arc's length and the full circumference relative to the subtended angle and the total 360-degree rotation. $$ 2 \pi : \alpha = 2 \pi r : L $$ Here, 2π represents the full rotation in radians, α is the subtended angle in radians, 2πr is the circumference's length, and L is the arc length. By rearranging the formula: $$ 2 \pi : \alpha = 2 \pi r : L $$ And reformulating it as fractions: $$ \frac{2 \pi }{ \alpha } = \frac{2 \pi r }{L } $$ Dividing both sides by 2π gives: $$ \frac{1 }{ \alpha } = \frac{ r }{L } $$ Multiplying both sides by L yields: $$ \frac{L }{ \alpha } = r $$ And multiplying both sides by α results in: $$ L = r \cdot \alpha $$ This is how the final formula to calculate the arc length based on the central angle is derived.

Example

Let's take a circle with radius r=3.

an example of a circle

A 45° central angle forms an arc between points A and B on the circle.

central angle of 45°

45° in radians is equivalent to π/4.

$$ \alpha = \frac{ \pi \cdot 45° }{180} = \frac{ \pi }{ 4 } $$

Thus, the length of the arc L from A to B, moving counterclockwise, is:

$$ L = \alpha \cdot r = \frac{ \pi }{4} \cdot 3 = 2.36 $$

The arc length for a 45° central angle is therefore 2.36.

the arc length

Further Observations

Additional insights on circle arcs include:

Equal arcs correspond to equal central angles and equal angles on the circumference.

  • Proportionality
    The arc's length is directly proportional to the circle's radius. For example, doubling the radius doubles the arc's length.

    Note. As an example, for a circle with radius r=3 and a central angle of 45° (π/4 radians), the arc's length is approximately 2.356, rounded here to 2.36. $$ L = \alpha \cdot r = \frac{ \pi }{4} \cdot 3 ≈ 2.356 $$
    the arc length
    Doubling the central angle to 90° (π/2 radians) roughly doubles the arc's length to about 4.71, which is twice 2.356. $$ L = \alpha \cdot r = \frac{ \pi }{2} \cdot 3 ≈ 4.71 $$
    the 90° arc

  • Chord
    The line segment connecting the endpoints of an arc, A and B, is called a chord. Each arc corresponds to a particular chord.
    an example of a chord
  • Circular Sector
    The area enclosed by an arc and the two radii connecting its endpoints to the circle's center is known as a circular sector.

    the circular sector

  • Circular Segment
    The area enclosed by an arc and its corresponding chord is known as a circular segment.

    the circular segment

And so forth.

 

 
 

Please feel free to point out any errors or typos, or share suggestions to improve these notes. English isn't my first language, so if you notice any mistakes, let me know, and I'll be sure to fix them.

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