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

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

Difficulty: Moderately Easy

Instructions

Things You'll Need:

• pencil
• paper
• scientific calculator

1. 

   This graphic of a pizza is a representation of a circle subtended by a central angle...

2. The RED LINE represents the ARC LENGTH, denoted by the letter 's'...

3. The BLACK LINES represent the RADIUS, denoted by the letter 'r'...

4. The GREEN LINE represents the DEGREE MEASURE OF THE CENTRAL ANGLE, denoted by the Greek letter theta...

5. The BLUE LINE represents the circle's CIRCUMFERENCE, denoted by the letter 'C'

6. 

   The EQUATION for calculating the ARC ANGLE RELATIONSHIP is theta/360° = s/C = s/2πr

7. By knowing any two of the three quantities: s, C, or theta, we can calculate the third using basic algebra.

8. EXAMPLE PROBLEM: Suppose you're asked to calculate the circumference of a pizza. C = ?; s = 4in; theta = 90°

9. Substitute the values into the arc angle relationship formula... EXAMPLE: (90°)/(360°) = 4/C

10. 

    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

11. 

    Multiply both sides by 4, and solve for C... EXAMPLE: (4)(360°)/(90°) = C/4(4); C = (4) * (360°)/(90°); C = 16in

12. EXAMPLE PROBLEM #2: Suppose you're asked to calculate the arc length of a slice of pizza. theta = 30°; r = 6in; s = ?

13. 

    Substitute the values into the arc angle relationship formula... EXAMPLE: (30°)/(360°) = s/2π6

14. 

    Multiply both sides by (2π6), and solve for s... EXAMPLE: (2π6) * (30°)/(360°) = s/2π6 * (2π6)

15. 

    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.