How to calculate the angle of repose

The angle of repose is the minimum angle at which any piled-up bulky or loose material will stand without falling downhill. One way to demonstrate this would be to pour sand from a bag to the ground. There is a minimum angle or maximum slope the sand will maintain due to the forces of gravity and the effect of friction between the particles of sand. The angle is calculated between the peak of the pile and the horizontal ground. The angle of repose for dry sand has been calculated to be 35 degrees, whereas cement has an angle of repose of 20 degrees.

Pour the dry sand into a pile on a level surface allowing it to build a pile from the top. This will result in a pile with a relatively circular base, making measurement easier.

Using the ruler and a tape measure, measure the height (h) of the pile of sand from the peak to the ground. Stand the ruler next to the pile so it can be read easily. Extend the tape measure carefully to the top of the pile without disturbing the pile and allow the other end of the tape measure to intersect the ruler. While keeping the tape measure level, observe the intersection of the tape measure with the ruler. Write the value on the paper. (Example: h = 12 inches.)

Using the tape measure, measure the horizontal distance (d) from the middle of the pile to the edge. Place the tape measure on the ground beside the pile. Line up one end with one side of the pile and extend the tape measure to the other end of the pile. Write the value on the paper and divide by 2. This will give you the distance from the centre of the pile to the edge. (Example: Total distance on tape measure from one end of the pile to the other = 30 inches. Divide by 2 to get 15 inches. d = 15 inches)

The equation for calculating the angle of repose is: tan-1(2h/d). Using your scientific calculator, multiply height (“h” calculated in Step 2) by 2 and divide this value by the distance (“d” calculated in Step 3). Then, hit the inverse tan key (or tan-1) and the answer just calculated. This will give you the angle of repose, α.

Place the protractor on the level surface next to the pile of sand. Using the ruler, create a straight line from the peak of the sand pile down the slope. Read the angle of repose value (α) and write the value on the paper.

Compare the calculated angle of repose from Step 4 and the measured angle of repose from Step 5. If the values are not within 1 degree of each other, repeat Step 5.