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How do you find the length of a circular path?

Published in Geometry 2 mins read

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).