Angles in a unit circle word problem (2)

Question

In Grade 10 Mathematics, you learned already that the radian measure of a central angle of a circle can be used to easily find the length of the intercepted arc of the circle. The length s of an intercepted arc by a central angle of θ radians on a circle of radius r is given by s=rθ . Use this learned concept to answer the following:

a) A unit circle (radius in cm) is divided into six congruent arcs. What would the length of each arc be? Express answer in terms of π .

b) On a circular balloon of radius 1 meter, the length of an arc is 1.742 meters. What angle in degree measure does it subtend at the center? π=3.14 .

c) A thread is winded on a spool of radius 1 inch. Points A, B, C, D, and so on up to I, are marked on the spool and equally spaced. What length of thread is used to wind up from point A to point E? π=3.14

Answer 1

A unit circle has a radius of 1 unit. In (a) it is 1 cm, in (b) it is 1 m, and in (c) it is 1 inch.

You have been given the formula for arc length: s = rθ

For a unit circle, r = 1, so the arc length is equal to the angle at the center (in radians!)

(a) Six congruent arcs means six arcs of equal length. This also means six angles of equal size. The total angle of a revolution is 2π, so each angle will be 2π/6 or π/3. Each arc length will therefore be π/3 cm.

(b) An arc length of 1.742 m subtends an angle in the center of 1.742 radians (because this is a unit circle). All you have to do now is to convert 1.742 radians into degrees. Remember π radians = 180 degrees.

(c) This circle is divided into 9 equal parts. You want to calculate the length of the arc AE, which is ⁴/₉ of the circumference. Since this is another unit circle, the total circumference will by 2π inches. What is ⁴/₉ of 2π?

Answer 2

The length s of an intercepted arc by a central angle of θ radians on a circle of radius r is given by s=rθ .

What prevents you from using this fact?