Firefighter Math: 4.5 Slope

Slope refers to the angle, or grade, of an incline. Slope can be upward or downward. Slope is typically expressed as a percent, and corresponds to the amount of rise, or vertical distance, divided by the run, or horizontal distance. Percentage means per 100. Slope can also be expressed as an angle, which gives the amount of deviation from flat as a number of degrees. Conversions between slope percent and slope angle can be done using a scientific calculator and the inverse tangent (arc tan) function. Essentially, the slope angle is the inverse tangent of the slope percent (with slope percent expressed in decimal).

Example 1 - The slope percent is 60 percent. What is the slope angle? 
Step 1. Change 60 percent to decimal form. Sixty percent means 60 out of 100. It can be written 60/100 = 0.60. See Chapter 1.
Slope angle = inverse tan of the slope percent (in decimal)
Slope angle = inverse tan of 0.60 
Step 2. Enter .6 into the calculator and push the inverse, inv, or "2nd" button, then the tan button to get the inverse tangent. The calculator will show the slope angle.

A 60 percent slope corresponds to a slope angle of 31°.

MEASURING SLOPE PERCENT

Slope percent can be measured using a clinometer or slope meter, or by dividing the rise by run, as described in this multimedia tutorial. Click the graphic below to view the lesson, which includes audio.


Click on the above graphic to view a slope measurement tutorial. 
 

If you have a clinometer or other digital device for measuring slope percent in the field, sight the clinometer as outlined below:
1. Open both eyes to sight the object and read the scale.
2. Verify which scale is being read. There are two scales in the viewfinder: a percent slope scale on the right margin and a slope angle scale on the left margin. The vertical angle is in degrees. 
3. Sight the clinometer from eye level to the object or to a distant point that is also at about eye level.
4. Read the scales for percent slope or degree of slope.
 
Note that in uneven terrain, the clinometer should be placed on a pole at eye level and read to a distant point on another pole of the same height to obtain a more accurate reading.

Example 2 - Use the rise and run measurements in the figure below to estimate the slope percent.
 

Slope percent = (8 feet / 40 feet) × 100 = 0.20 × 100 = 20%

The slope percent is 20 percent.

CALCULATING HORIZONTAL DISTANCE

If the slope and the vertical distance (rise) are known, then the horizontal distance (run) can be calculated. The slope percent equation can be rearranged to provide the equation for the horizontal distance.

Slope percent = (rise / run) × 100
Rearrange the terms of equation: multiply both sides by run.
run × slope % = rise/run × 100 × run
Divide both sides by slope percent.
( run × slope %) / (slope %) = (rise × 100) / (slope %)

run = (rise × 100 ) / slope % is a measure of horizontal distance.

Example 3 - A hill has a slope of 8 percent. The height of the hill is 15 feet. What is the horizontal distance?

horizontal distance = run = (rise × 100) / slope %

Step 1. Enter the given values into the equation.

Step 2. Solve.
run = ((15 ft × 100) / 8) = (1500 ft / 8)= 188 ft
 

The hill has a horizontal distance of 188 feet.

CALCULATING SLOPE DISTANCE

Slope distance (h) is the length of slope from the bottom to the top of the slope and is larger than both the vertical and horizontal distance. 

Slope distance can be calculated when the vertical height (rise) and the horizontal distance (run) of a right angle are known. There is a right angle if the vertical and horizontal distances are "true" to the vertical and horizontal, respectively. See the following figure, which denotes x as run and y as rise. To calculate slope distance, you will need a basic scientific calculator with a square root (√z ) function.
 

Example 4 - Find the slope distance for the vertical and horizontal distances illustrated in the figure below.


Step 1. Use the equation h = √(x²+ y²) 
slope distance = 
    √ [(horizontal distance)² + (vertical distance)²]

Step 2. Change all the values to the same units, in this case feet. The conversion factor is 12 inches = 1 foot.



Step 3. Plug the values into the equation and solve.
h = √ (x² + y²)

h = √[(41.7 ft × 41.7 ft) + (9.3 ft × 9.3 ft)] = √ [(1738.9 ft² + 86.5ft²)] 

h =√ (1825 ft²) = 42.7 ft

What is the slope distance in feet and inches?
h = 42 ft + 0.7 ft × 12 in/1 ft = 42 ft 8 in

See Chapter 2, Section 2.1 for a review of unit conversions.

h = slope distance = 42.7 ft or 42 ft 8 in