Calculating angles for making shelf supports, which are based on right-angled triangles, will take you back to high school math class. Whether you look on it with dread or delight, the Pythagorean theorem has real-world applications, and calculating shelf support angles is one of them. If it has been a while since you racked your brain over trigonometry, you may opt for a protractor. Otherwise, you can calculate shelf support angles.
Measure the distance down the wall you want the support to go, as well as the distance under the shelf you want the support to extend to. For example, say the wall measurement, side "a" is 12 inches and the shelf measurement side "b" is 8. The angle between the wall and shelf measurement, angle "C," we know will be 90 degrees.
Find the distance of the hypotenuse, side "c" created between the end of the wall and shelf measurements by using the Pythagorean theorem. The Pythagorean theorem is "a" squared plus "b" squared is equal to "c" squared. In the example this would be: 12 squared plus 8 squared equals c squared, or 144 + 64 = 208 squared. The square root of 208 will be the measurement for "side c." So "c" equals 14.42 inches.
Find the two missing angles of the shelf support (angles "A" and "B") using the inverse trigonometric functions for sine, cosine and tangents. For example, to find angle "A," use the measurements for sides "b" and "c," which are adjacent to angle "A." Cos A = b/c. So in the example, cos A = 8/14.42, or 0.55.
Enter the arccosine of 0.55 into a calculator to get the angle "A" number. For the example, arccosine of 0.55 = 56.3, so A = 56.31.
Repeat for angle "B" using cos B = a/c. For the example, this would be B = 12/14.42 which equals 0.83. Enter the arccosine of 0.83 into a calculator to get angle "B." Arccosine 0.83 = 33.69. So you have found all the angles for the shelf support, with angle "c" equaling 90 degrees, angle "b" equaling 33.69 and angle "c" equaling 56.31 degrees.