Arc of a Circle
A circle arc is a section of the circumference defined by two points, A and B.
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.
A 45° central angle forms an arc between points A and B on the circle.
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.
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. $$ L = \alpha \cdot r = \frac{ \pi }{4} \cdot 3 ≈ 2.356 $$
Doubling the central angle to 90° (π/2 radians) roughly doubles the arc's length to about 4.71, which is twice 2.356. $$ L = \alpha \cdot r = \frac{ \pi }{2} \cdot 3 ≈ 4.71 $$
- Chord
The line segment connecting the endpoints of an arc, A and B, is called a chord. Each arc corresponds to a particular 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.
- Circular Segment
The area enclosed by an arc and its corresponding chord is known as a circular segment.
And so forth.