Length of an Arc (Radians)
(KS3, Year 7)
The LessonThe length of an arc of a circle is given by the formula:
In this formula, r is the radius of the circle and θ is the angle (in radians) subtended by the arc. The image below shows what we mean by the length of an arc:
How to Find the Length of an Arc of a Circle (Radians)Finding the length of an arc of a circle, when the angle is in radians, is easy.
QuestionWhat is the length of the arc with an angle of 1 radian and a radius of 5 cm, as shown below?
Start with the formula:
Length of arc = rθ
Substitute the radius and the angle into the formula. In our example, r = 5 and θ = 1.
Length of arc = 5 × 1 Length of arc = 5 cm
Answer:The length of an arc of a circle with a radius of 5 cm, which is subtended by an angle of 1 radian, is 5 cm.
Lesson SlidesThe slider below shows another real example of how to find the length of an arc of a circle when the angle is in radians. Open the slider in a new tab
What Is an Arc?An arc is a portion of the circumference.
What Are Radians?Radians are a way of measuring angles. 1 radian is the angle found when the radius is wrapped around the circle.
Why Does the Formula Work?An angle in radians can be found using the formula:
Angle = Arc length ⁄ RadiusWe can rearrange this formula to make the arc length the subject:
Arc length = Angle × Radius Arc length = θ × r
Is the Angle Given in Degrees or RadiansThe formula to find the length of an arc of a circle depends on whether the angle at the center of the arc is given in degrees or radians. Make sure you check what units the angle is given in.
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