How to Measure Acute Angles Without a Protractor

You don't need a protractor to measure an acute angle.

The protractor is a tool that directly measures angles, but it is not something that is commonly carried around in one's pocket. If you find yourself in a situation that requires you to measure an acute angle in a pinch, however, you can do so with a ruler and a pocket calculator using fundamental trigonometry. The term "SOH CAH TOA" is the mnemonic device used to remember which ratios to measure when using the sine, cosine and tangent functions, which are used to determine angles. "SOH" stands for "sine equals opposite over hypotenuse; "CAH" means "cosine equals adjacent over hypotenuse"; and "TOA" stands for "tangent equals opposite over adjacent."

Instructions

1. Make a right angle with one of the arms of the acute angle, and continue the other arm of the right angle until it intersects with the second arm of the acute angle.

2. Measure the length of the side of the triangle from the acute angle to the right angle. This is called the adjacent arm.

3. Measure the length of the other arm from the acute angle. This is called the hypotenuse.

4. Divide the length of the adjacent side by the length of the hypotenuse.

5. Evaluate the inverse cosine of the ratio of the adjacent arm to the hypotenuse with a scientific calculator. This will return the value of the angle in degrees. You can convert the angle from degrees to radians using a trigonometric table.

6. Locate the value of the cosine of the angle in a trigonometric table (see Resources), and the corresponding value in degrees or radians will be listed. Use this method if you do not have a calculator available.

References

• The Math Page: Trigonometry of Right Angles
• Math Words: SOHCAHTOA

Resources

• The Math Page: Three-Place Trigonometric Table