How to make a protractor.

1. Per Hand method 1
Draw a square of 11.5cm sides. (Or of course any multiple.
And of course any other decimal base measurement can be substituted for cm.)
Draw a circle of 11.5cm diameter exactly on the square.
Draw the 45°, 90°, 135° etc diagonals in using the square.
Now 20 and 10 and 5 degree marks may be made - 20° points are 2cm apart on 
the circle (measured as a line between the 2 points, and not as an arc on 
the circle). 10° - 1cm, 5° - 0.5cm.
1° points are now made, 1mm apart.
Cut out the circle/half circle/quarter circle. 

Accuracy:
1° 0.1004cm
5° 0.5016cm
10° 1.0023cm
20° 1.997cm
45° should measure 4.4009cm
90° should measure 8.132cm

This works because the sine is nearly linear for small values.


2. Per Hand method 2, with graph paper: protractor.gif

The errors in the angle using the given approximations are:
angle   per cent error    approximation used
 5        -0.605            2/23
10         0.080            3/17
15         0.122           11/41
20        -0.084            4/11
25         0.068            7/15
30        -0.061           15/26
35        -0.023            7/10
40        -0.033           26/31