How to Calculate the Arc Length, Central Angle, and Circumference of a Circle

By chrisjenkins

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Learn how to calculate the arc length, central angle, and circumference of a circle.

Instructions

Difficulty: Moderately Easy

Things You’ll Need:

  • pencil
  • paper
  • scientific calculator

Step1
This graphic of a pizza is a representation of a circle subtended by a central angle...
Step2
The RED LINE represents the ARC LENGTH, denoted by the letter 's'...
Step3
The BLACK LINES represent the RADIUS, denoted by the letter 'r'...
Step4
The GREEN LINE represents the DEGREE MEASURE OF THE CENTRAL ANGLE, denoted by the Greek letter theta...
Step5
The BLUE LINE represents the circle's CIRCUMFERENCE, denoted by the letter 'C'
Step6
The EQUATION for calculating the ARC ANGLE RELATIONSHIP is theta/360° = s/C = s/2πr
Step7
By knowing any two of the three quantities: s, C, or theta, we can calculate the third using basic algebra.
Step8
EXAMPLE PROBLEM: Suppose you're asked to calculate the circumference of a pizza. C = ?; s = 4in; theta = 90°
Step9
Substitute the values into the arc angle relationship formula... EXAMPLE: (90°)/(360°) = 4/C
Step10
It is easier to isolate C if it is in the numerator, so take the reciprocal of both sides (flip the numerator *top* with the denominator *bottom*)... EXAMPLE: (360°)/(90°) = C/4
Step11
Multiply both sides by 4, and solve for C... EXAMPLE: (4)(360°)/(90°) = C/4(4); C = (4) * (360°)/(90°); C = 16in
Step12
EXAMPLE PROBLEM #2: Suppose you're asked to calculate the arc length of a slice of pizza. theta = 30°; r = 6in; s = ?
Step13
Substitute the values into the arc angle relationship formula... EXAMPLE: (30°)/(360°) = s/2π6
Step14
Multiply both sides by (2π6), and solve for s... EXAMPLE: (2π6) * (30°)/(360°) = s/2π6 * (2π6)
Step15
EXAMPLE: (2π6) (30°)/(360°) = s; s = 3.14in

Tips & Warnings

  • Remember, if you're given the diameter instead of the radius simply divide the diameter by 2.

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eHow Article: How to Calculate the Arc Length, Central Angle, and Circumference of a Circle

eHow Member: chrisjenkins

chrisjenkins

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