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Gradient is rise over run in the same unit, written as one in so many or as a percent grade, and worked from a map's contours and scale.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to work out a gradient from a map, write it as one in so many and as a percent, and judge how steep it is.
You can read heights from contour lines and turn a map distance into a ground distance with the scale. Gradient combines the two: how much the land rises for each meter you travel across it.
| Term | What it means |
|---|---|
| Gradient | The steepness of a slope: the rise divided by the horizontal distance. |
| Rise | The difference in height between two points. |
| Run | The horizontal distance between two points, measured on the ground. |
| One in so many | A gradient written as one unit up for so many along, such as 1 in 20. |
| Percent grade | A gradient written as meters up for every hundred meters along. |
The gradient between two points compares how far up with how far along:
$$\text{gradient} = \dfrac{\text{rise}}{\text{run}}.$$
A bigger number after one in means a gentler slope.
Another way: picture
Picture walking up a ramp. If you climb one step for every twenty paces, the ramp is one in twenty. If you climb one step for every five paces, it is one in five, much steeper. The number after one in is how far you walk for each unit of height.
Another way: steps
A slope that rises $50$ m over $1000$ m can be written three ways.
| Form | Working | Value |
|---|---|---|
| fraction | 50 ÷ 1000 | 0.05 |
| one in so many | 1000 ÷ 50 | 1 in 20 |
| percent grade | 0.05 × 100 | 5% |
All three describe the same slope. Hikers and geographers often use one in so many; road engineers use percent grade.
The run is horizontal distance on the ground, not distance on the paper and not distance along the slope. On a map where $1$ cm stands for $250$ m, a line $4$ cm long is a run of $1000$ m.
The distance you would walk up a slope is a little longer than the run, but for the gentle slopes most maps show the difference is tiny. Gradient always uses the horizontal run.
Read the height of each end from the contours, estimating between lines if needed. A trail from the $120$ m contour to the $170$ m contour rises $50$ m.
If one end is a spot height, use that number exactly. If the two ends are on the same contour, the rise is zero and the gradient is flat, however far apart they are.
A gradient has no unit, because it divides a length by a length. That only works when both are in the same unit. A rise in meters over a run in centimeters of map, or a drop in feet over a length in miles, gives a meaningless number.
Convert first: centimeters of map to meters of ground, miles to feet. Then divide.
A gradient of one in a hundred is nearly flat: a railroad or a canal towpath. One in twenty is a noticeable climb on foot. One in five is steep for a road and hard work for a hiker. One in one is a slope of forty-five degrees, a scramble.
Percent grades read the other way: a bigger percent means a steeper slope. A $5$ percent grade is one in twenty; a $20$ percent grade is one in five.
On a steady slope of one in $20$, every $20$ m along gains $1$ m of height. After $x$ meters along, the height gained is $x \div 20$. That is a straight-line function: double the distance and you double the height gained.
Real slopes change gradient as they go, which is why a map shows contours rather than one number, but over short stretches a steady gradient is a good model.
Gradient decides where roads and railroads can go. Trains struggle on grades above about two percent, so railroads wind around hills and through tunnels to keep their slopes gentle.
Rivers flow faster on steep gradients and erode their beds more; on gentle gradients they slow, drop sediment and wander. Farmers terrace steep slopes to stop soil washing away. Gradient is one of the most useful numbers a map can give.
Mountain highways in the United States post warning signs such as 6% grade next 5 miles. Long steep grades are dangerous for heavy trucks, whose brakes can overheat on the way down, so many mountain highways have runaway truck ramps, steep gravel lanes that stop a truck whose brakes have failed.
Interstate highways are usually designed to keep their grades to about six percent or less even in mountains. City streets can be much steeper: some streets in San Francisco climb at grades over thirty percent.
Checking an answer. One in so many should be bigger than one for any ordinary slope. If you get a number smaller than one, you divided the wrong way.
Dividing rise by run is allowed because it measures height gained per unit of horizontal distance, which is what steepness means. Converting units first is required because a ratio only makes sense when both parts measure the same thing in the same unit.
Writing the same slope as a fraction, a ratio or a percent is allowed because each is the same number written differently.
A cross-section, the subject of the next lesson, draws the shape of the land along a line. On it, gradient is visible as the slope of the line: steeper stretches rise faster.
Drawing a cross-section to the same scale up and across shows gradients truthfully, but most cross-sections stretch the height to make small slopes visible, which makes every slope look steeper than it is.
The most common slip is turning the ratio upside down and writing twenty in one instead of one in twenty. Another is dividing by the map distance in centimeters instead of the ground distance in meters.
A third is mixing units, such as feet over miles. A fourth is reading a bigger one-in number as steeper, when it is gentler.
Without calculating, you can judge gradient from contour spacing. If the interval is $20$ m and neighboring contours are $1$ cm apart on a map at $1$ cm to $250$ m, the slope is one in about $12$; if they are $2$ cm apart, one in about $25$.
That quick estimate tells a hiker where the hard stretches are before they set out.
On Interstate 70 west of Denver, Colorado, the highway descends from the Eisenhower Tunnel toward the town of Silverthorne and later climbs over Vail Pass, with long stretches of grades around six and seven percent. Signs warn truck drivers to use low gears, and gravel runaway truck ramps wait beside the road.
A loaded truck going down a long grade relies on its brakes and engine to hold its speed. If the driver rides the brakes, they can overheat and fail, and the truck speeds up. A runaway ramp is a lane of deep loose gravel, often climbing uphill, that slows the truck safely.
The percent on the sign comes straight from rise over run: the drop in feet divided by the length of road in feet, times one hundred. Engineers use the same arithmetic when they design a new road, choosing a route that keeps the grade low enough for trucks to climb and descend safely.
San Francisco was laid out in a grid of straight streets in the 1800s, drawn without much thought for the hills underneath. The result is some of the steepest city streets in the United States. Filbert Street between Leavenworth and Hyde rises at a grade of about thirty-one percent, and a few other blocks are nearly as steep.
A thirty-one percent grade means about thirty-one feet up for every hundred feet along, roughly one in three. Cars park sideways with their wheels turned toward the curb, and some sidewalks become staircases.
The city's famous cable cars were invented in the 1870s partly to climb such grades, which were too steep for horses pulling streetcars. A cable running under the street pulls the car up the hill, however steep. The contour map of the city shows exactly where the grid runs straight up the slopes, with the lines packed tightly together.
It is easy to write the numbers the wrong way around, giving twenty in one for a gentle slope that is really one in twenty, or to divide by the length of the line on the map in centimeters. Both give numbers that describe no real slope.
Always turn the map distance into ground distance first, check that the rise and run are in the same unit, and remember that one in so many is run over rise: a gentle slope has a big number after the one in.
A path rises $40$ m over $600$ m. Write rise over run.
$\dfrac{40}{600}$
Height over distance.
Divide the run by the rise.
$\dfrac{600}{40} = 15$
Meters along per meter up.
Write the gradient.
$1 \text{ in } 15$
One up for fifteen along.
Write it as a percent.
$\dfrac{40}{600} \times 100 \approx 6.7\%$
Rise over run times one hundred.
A trail runs $6$ cm on a map where $1$ cm is $250$ m. Find the run.
$6 \times 250 = 1500\ \text{m}$
Ground distance.
It climbs from the $300$ m contour to the $400$ m contour. Find the rise.
$400 - 300 = 100\ \text{m}$
Height gained.
Divide the run by the rise.
$\dfrac{1500}{100} = 15$
One in fifteen.
Write the percent grade.
$\dfrac{100}{1500} \times 100 \approx 6.7\%$
The same slope.
Judge the slope.
$\text{a steady climb on foot}$
Steeper than most roads.
A highway drops $1584$ ft over $5$ miles. Convert the length to feet.
$5 \times 5280 = 26400\ \text{ft}$
Five thousand two hundred eighty feet per mile.
Divide the drop by the length.
$\dfrac{1584}{26400} = 0.06$
Rise over run.
Write the percent grade.
$0.06 \times 100 = 6\%$
The sign's number.
Write it as one in so many.
$\dfrac{26400}{1584} \approx 16.7$
About one in seventeen.
Find the drop per mile.
$\dfrac{1584}{5} \approx 317\ \text{ft}$
A useful check.
Say why trucks slow down.
$\text{brakes overheat on long grades}$
Hence the warning sign.
Write the run over the rise.
$\dfrac{500}{25}$
Meters along per meter up.
Evaluate the expression.
$20$
One in twenty.
Write the percent grade.
A path rises $25$ m over a horizontal distance of $500$ m. What is its gradient, written as one in so many?
Complete the worked solution: on a map where $1$ cm stands for $250$ m, a stream flows $4$ cm from a spring on the $170$ m contour down to a lake at $120$ m. Find the run, the drop and the stream's gradient as one in so many.
Find the run on the ground.
$\text{map cm} \times 250 =$ h
Meters along.
Find the drop.
$\text{spring} - \text{lake} =$ r
Meters down.
Divide the run by the drop.
$\dfrac{\text{run}}{\text{drop}} =$ n
One in so many.
Say what a steeper stream does.
$\text{flows faster, erodes more}$
The next unit, on rivers.
Match each rise and horizontal distance to its gradient.
| 1 in 20 | 1 in 5 | 1 in 100 | 1 in 10 | |
|---|---|---|---|---|
| a rise of 30 m over 600 m | ||||
| a rise of 50 m over 250 m | ||||
| a rise of 10 m over 1000 m | ||||
| a rise of 40 m over 400 m |
On a map where $1$ cm stands for $250$ m, a trail runs $8$ cm from a point on the $220$ m contour to a point on the $300$ m contour. Fill in the run, the rise and the gradient as one in so many.
| value | |
|---|---|
| run on the ground (m) | |
| rise (m) | |
| gradient, 1 in |
A railroad climbs a steady slope of one in $8$. Write the height it gains, in meters, as a function of the horizontal distance $x$ it travels, in meters.
Answer:
A road rises $45$ m over a horizontal distance of $900$ m. What is its grade, as a percent?
Answer: %
Suppose a mountain highway in Tennessee drops $1056$ feet over $4$ miles, which is $21120$ feet of horizontal distance. What percent grade should the warning sign for truck drivers show?
Answer: %
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A railroad climbs a steady slope of one in $50$. Write the height it gains, in meters, as a function of the horizontal distance $x$ it travels, in meters.
Answer:
You can find a gradient from a map. Explain why the map distance must be turned into ground distance before you divide.
24. Your turn: a slope rises $25$ m over $500$ m. What is its gradient as one in so many?, step 3
$\dfrac{25}{500} \times 100 = 5\%$
The same slope.