Length of an Arc
(KS3, Year 7)
The LessonThe length of an arc of a circle is given by the formula:
In this formula, θ is the angle (in degrees) subtended by the arc and r is the radius of the circle. The image below shows what we mean by the length of an arc:
How to Find the Length of an Arc of a CircleFinding the length of an arc of a circle is easy.
QuestionWhat is the length of the arc with an angle of 60° and a radius of 5 cm, as shown below?
Start with the formula:
Length of arc = θ⁄360° × 2πrDon't forget: π is pi (≈ 3.14) and / means ÷.
Substitute the angle and the radius into the formula. In our example, θ = 60° and r = 5.
Length of arc = 60°⁄360° × 2 × π × 5 Length of arc = (60° ÷ 360°) × 2 × 5 × π Length of arc = 5.2 cm
Answer:The length of an arc of a circle with a radius of 5 cm, which is subtended by an angle of 60°, is 5.3 cm.
Lesson SlidesThe slider below shows another real example of how to find the length of an arc of a circle. Open the slider in a new tab
What Is an Arc?An arc is a portion of the circumference.
Why Does the Formula Work?The length of an arc is just a fraction of the circumference of the circle of the same radius. The circumference is given by 2πr, where r is the radius. For example, an arc that is halfway round a circle is half the circumference of a circle.
An arc that is a quarter way round a circle is quarter the circumference of a circle.
In each case, the fraction is the angle of the arc divided by the full angle of the circle.
When measured in degrees, the full angle is 360°. Hence for a general angle θ, the formula is the fraction of the angle θ over the full angle 360° multiplied by the circumferece of the circle:
Length of arc = θ⁄360° × 2πr
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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