The length of a circular path, also known as the circumference or arc length, depends on whether you're considering the full circle or just a portion of it. Here's how to calculate it:
1. Finding the Circumference of a Full Circle
The circumference (C) of a circle, which represents the length of the entire circular path, is calculated using the following formulas:
- C = 2πr where r is the radius of the circle.
- C = πd where d is the diameter of the circle (and d = 2r).
- π (pi) is a mathematical constant approximately equal to 3.14159.
Example:
If a circle has a radius of 5 cm, its circumference is:
C = 2 π 5 cm = 10π cm ≈ 31.42 cm
2. Finding the Arc Length of a Portion of a Circle
The arc length (s) is the length of a specific segment of the circle's circumference, defined by a central angle. The formulas depend on whether the angle (θ) is given in radians or degrees:
-
If θ is in radians:
s = rθ where r is the radius and θ is the central angle in radians.
-
If θ is in degrees:
s = (πrθ) / 180 where r is the radius and θ is the central angle in degrees. This formula essentially converts the angle from degrees to radians before applying the s=rθ formula.
Example 1 (Radians):
A circle has a radius of 8 cm and a central angle of π/4 radians. The arc length is:
s = 8 cm * (π/4) = 2π cm ≈ 6.28 cm
Example 2 (Degrees):
A circle has a radius of 10 cm and a central angle of 60 degrees. The arc length is:
s = (π 10 cm 60) / 180 = (10π)/3 cm ≈ 10.47 cm
Summary Table
| Scenario | Formula | Variables |
|---|---|---|
| Full Circle (Circumference) | C = 2πr or C = πd | r = radius, d = diameter |
| Arc Length (Radians) | s = rθ | r = radius, θ = angle |
| Arc Length (Degrees) | s = (πrθ) / 180 | r = radius, θ = angle |
In summary, to find the length of a circular path, you either calculate the full circumference using the radius or diameter or determine the arc length using the radius and the central angle (in radians or degrees).
