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$$ S = r \theta $$ Formula and Equation

Radian Measure and Arc of a Circle

There is a formula that relates the arc length of a circle of radius, r, to the central angle, $$ \theta$$ in radians.

Formula for $$ S = r \theta $$

The picture below illustrates the relationship between the radius, and the central angle in radians. The formula is $$ S = r \theta $$ where s represents the arc length, $$ S = r \theta$$ represents the central angle in radians and r is the length of the radius.

Picture of s = rTheta in a circle

Demonstration of the Formula $$ S = r \theta$$

The interative demonstration below illustrates the relationship between the central angle of a circle, measured in radians, and the length of the intercepted arc. Just click and drag the points.

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Practice Problem

Problem 1

What is the value of the arc length S in the circle pictured below?

illustration

Use the $$ S = r \theta $$ formula.

S = 3(Π/3) = Π.

Problem 2

What is the value of the arc length S in the circle pictured below?

illustration

Use the formula for S = r Θ and calculate the intercepted arc: .

Problem 3

Calculate the measure of the arc length S in the circle pictured below?

illustration

Use the formula for S = r Θ and calculate the intercepted arc: .

Problem 4

Calculate the measure of the arc length S in the circle pictured below?

illustration

Use the formula for S = r Θ and calculate the solution.


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