IB Class Notes

Radians & Degrees

Aim: By the end of this note, you will be able to transform degrees to radians and vice versa.

  1. You probably tell me the answer is 360o (degrees). Is this the only unit for measuring angle? Before we answer that question, let us first look into the history of why a circle has 360o (degrees).
  2. The answer lies with the ancient Babylonians. They used a base 60 number system. Part of its legacy is still with us. We say 1 minute = 60 seconds and 60 minutes = 1 hours.
    In notations these are 3600 '' (seconds) = 60 ' (minutes) = 1 o (degree in this context).
  3. The ancient Babylonians also had 360 days in a year. So every 360 days, they "experienced" a cylce or rotation (winter-spring-summer-autumn-winter). Thus, a circle is logically defined as a rotation that has 360o (degree). A degree is then defined as 1/360 of a full rotation. But this definition is not very satisfactory to mathematicians. Instead of 360 parts, why not divide a rotation into 100 parts or even 400 parts? They would like to have a definition that depends only on the characteristics of a circle.
  4. Here is that attempt.

    A radian is defined as the size of the angle such that the length of the arc, part of the circumference that belongs to sector \( \theta \), is the same as the radius. Sector is the shaded area in this diagram.

    \[ \begin{align} \large 1^c = \frac{180^o}{\pi} \\ \large \pi^c = 180^o \end{align} \]

Why is 1 c = 180o/π?

We know that the circumference of a complete circle (\(C\)) is 2πr. The arc length (\(s\)) of a sector with angle θ is thus
\( s = (\frac{\theta}{360^o})2\pi r \)
If θ = 1 radian then \( s = r \).
Thus we have, \( r = (\frac{1 radian}{360^o})2\pi r \)
\( 1 = (\frac{1 radian}{180^o})\pi \)
\( 1 radian = \frac{180^o}{\pi} \)
\( 1 radian \approx 57.3^o \) ( 3 s.f.)

Example 1.Convert the following angles to radians.
(i) 85o12'18"
(ii) 260o12"
Solutions:
(i) 85o12'18" = [ 85+ (12/60) + (18/3600) ]o ;
we first change the whole angle into a decimal expression in degree
85.205o = 85.205(π/180o) radian
≈ 1.49 radian (3 s.f.)

(ii) 260o15" = [ 260 + (15/3600) ]o(π/180o) radian
≈ 4.54 radian (3 s.f)
Example 2.Express the following angles in terms of degrees and minutes:
(i) 2.53c
(ii) 5(π/3) radian
Solutions:
(i) 2.53c = 2.53c(180oc)
= 144.9583222o (from calculator)
≈144o57' [to the nearest minute]
(ii) 5(π/3) radian = 5(π/3)c (180oc)
= 5(180o)/3
= 300o

Summary:

From θ radian to degree:   \( \theta^c \left( \frac{180^o}{\pi^c} \right) \)
From θ degree to radian:   \( \theta^o \left( \frac{\pi^c}{180^o} \right) \)

A way to remember the radian-degree conversion formula.

Let us study the two formula above. θ represents the angle that you need to convert, so it must be in the formula. After that we have 180o and πc to fit into our formula.


If we need the answer to be in degree then we just have to make sure 180o in the numerator and π will automatically go to the denominator.

If we need the answer to be in radian then we will have πc in the numerator and 180o in the denominator.

Worksheet

Unless otherwise specified in the question, report your answers exactly or to 3 significant figures.

  1. Complete the table below.
    0o

     

     
    60o
    180o
     
    270o 
     30o
     
    576o
    360o
    0c

     

    πc/8
     
    πc
     2πc/3
     
     
    πc/16
     
    c

  2. Convert the following angles to radian:
    1. 45o
    2. 17o 18'
    3. 164o 2' 30"

  3. Convert the following angles to degree:
    1. πc/7
    2. 0.59c
    3. 4.3c

  4. Assume that we have a circular pizza as in figure 1.
    Let the radius of this pizza be 10 cm and θ = 0.8 radian.
    1. Convert the angle θ = 0.8 radian to degree. Report your answer to 2 decimal places.

       

    2. How large is the angle θ as a percentage of the total circular pizza?

       

    3. You are given the sector AOB to enjoy and you especially love the crispy rim of pizza.
      Use your result in (b) above or otherwise to find the amount of crispy rim ACB that you can enjoy? Report your answer to 1 significant figure.
      [Hint: Arc ACB is a part of the circumference 2πr.]

       

       

       

      (Generalization)

    4. Instead of θ =0.8 radian, we will now have a general angle θ = xo.
      Find an expression for arc ACB in term of xo, π and r.

       

       

       

       

    5. Instead of measuring θ in degree we now measure the general angle θ in radian. Let this angle be θc. Find an expression for arc ACB in term of θc, π and r.

       

       

       

       

    6. Compare your answers for (d) and (e), and decide which expression for arc ACB is more simple?

       

  5. Let us assume that Hong Kong (H) and Kaifeng (K), China are located on the same longitude as in figure 2. Hong Kong is located on northern latitude 22o 15' and Kaifeng is on northern latitude 34o 36'. We further assume that the earth is a sphere with a radius of 6378 km. Let O be the center of a spherical earth and OE a part of the equator (i.e. latitude 0o).
    1. Find the angle θ in degree accurate to 2 decimal places. [Hint: the angle between HO and OE is 22o 15'.]

       

       

    2. Convert your answer in (a) to radian accurate to 3 decimal places.

       

       

    3. Use your result in (b) or otherwise to calculate the distance from Hong Kong to Kaifeng to the nearest km. [Hint: refer to your answers for question 4 above.]

       

       

       

       

Answers:
1) 22.5o, πc/3, 120o, 3pc/2, pc/6, 11.25o, 3.2pc (2a)πc/4 = 0.785 rad (2b) 0.302 radian (3c) 2.86 radian (3a) 25.7o (3b) 33.8o (3c) 246o (4a) 45.84o (4b) 12.7% (4c) 8 (4d) (x/360)2πr (4e) θcr (4f) the one with radian is more simple. (5a) 12.35o (5b) 0.216 radian (5c) 1378 km [1375 km if you use 12.35o(π/180o)(6378)]

 

Email KokMing Lee
Li Po Chun United World College of Hong Kong,
10 Lok Wo Sha Lane, Sai Sha Road,
Shatin, New Territories, Hong Kong SAR.