How to Find an Angle Without a Protractor

The sine, cosine and tangent functions relate the side lengths of a right triangle to its angles.

Finding the exact measure of an angle requires the use of a protractor or other mathematical instrument, but you can use formulas from trigonometry to get a very close approximation of the angle. The sine of an angle of a right triangle is the ratio of its opposite side to the hypotenuse of the triangle. Construct a right triangle from the angle and use the inverse sine function on your calculator to find the measure of the angle.

Things You'll Need

• Ruler
• Graphing calculator

Instructions

1. Label the angle you wish to measure P. Use your ruler to measure a certain number of inches along one side of the angle. Label the distance x and the point x inches away Q.

2. Measure the same number of inches along the other side of the angle and label the point x inches away R.

3. Use your ruler to draw a perfectly straight line connecting points Q and R. Measure half the distance from Q to R and label this point S. Label the distance between Q and S as y inches.

4. Divide y by x. For example, if the distance x were 8 inches and the distance y were 4 inches, you would divide 8 by 4 to get 2.

5. Use your calculator to find the arcsine of value you obtained in Step 4 to get the value corresponding to half the measure of angle P. The arcsine of y divided by x is equal to the measure of angle QPS, because QPS is a right triangle and y and x are the opposite and hypotenuse of angle P. The arcsine of this ratio gives the measure of the angle.

6. Double the result you obtained in Step 5 to get the measure of angle P. The measure of angle QPS is half the measure of the original angle P because the line connecting P and S is a bisector of the angle. Doubling the measure of QPS gives you the measure of angle P.

References

• New Mexico State University: How to Measure an Angle Without a Protractor