How To Calculate Angles Without A Protractor

A protractor is required to directly calculate the measure of an angle, but you can use geometric properties of triangles to make an indirect measure of the angle. Use the sine formula to infer the measure of the angle from the distance between two points along the angle's lines a certain distance from the angle's origin.

1. Mark Two Points on the Line Opposite the Angle

Use the ruler to measure a specific distance along both sides of the angle from the angle's origin (the same distance along both sides), and label this distance "d." Mark the two points on the angle that are "d" length away from the origin.

2. Measure the Line

Use the ruler to measure the exact distance between the two points on the angle. Label this distance "e."

3. Use the Sine Formula

Input the values of d and e into the formula "Angle measure = 2 x arcsin (0.5 x e/d)." (In other words, the angle measure is equal to two times the inverse sine of half the ratio between the lengths e and d.) This formula is derived from the equation for the sine given the measures of a right triangle's sides: The sine of the angle is equal to the length of the side opposite the angle divided by the length of the triangle's hypotenuse.

4. Calculate the Angle

Use your graphing calculator to solve for the angle measure. Type "2," then the multiplication symbol, "Arcsin" and the value that is half of e divided by d. Then hit "Enter" or "=" to see the answer. You may have to use the calculator's "2nd" key to enter "Arcsin." (It's usually located on the same key as sin.)