How to measure an angle without a protractor
Calculator. TI-34 II, or another scientific calculator.
Materials and tools.
Each student is given an angle drawn on a plain sheet of paper, ruler, compass, scientific calculator, but no protractor.
Task.
Measure the angle in degrees. Write the description of the procedure, the data you recorded, the result, and the mathematical facts that justify it.
Example of a solution.
A given angle.
Procedure.
On the rays of
the angle, mark two points at the same distance from the vertex. (Use either a compass or a ruler to do so.) Measure their
distance d from the vertex, and the distance e between them. On the TI-34 II
calculator, set
Data and result. (An example)
d = 8.7 cm
e = 2.7 cm
the_angle = 36.2 degrees
Answer: The angle is 36 degrees.
Facts used.
In an isosceles triangle, the height bisects its base, and also it bisects the angle opposite to it. So the top half of the triangle shown above is a right triangle with hypotenuse d and leg e/2. The sine of angle A, is sin(A) = .5e/d. But A is one half of the angle we have to measure, and therefore, the angle = 2*arcsin(0.5e/d).