How do I work out how steep the land will be and its effects?




Slope steepness is the proper term given to how steep the land is. To work it out, we need to count the amount of contour lines over a certain length and consider the scale of the map. (You can read about map scales and contour lines in the lessons: How do I measure distances? and How do I work out how high and steep the land is?) We express slope steepness as a gradient or an angle and with those measurements we can make a good guess about a section of the landscape we are going to hike in.

Understanding slope steepness and its effects allows us to:

Important notes


Slope steepness as a gradient or an angle


Gradient

Gradient is the the amount of vertical units ascended divided by the amount of horizontal units traveled forward. It is expressed as a percentage (%). If you climb 100 metres and travel 100 metres forward then the gradient of the slope is 100% (i.e 1 metre vertically for every 1 metre horizontally). So if you climb 50 metres up and 100 metres along then the gradient will be 50% but if you climb 100 metres up and 50 metres forward then the gradient will be 200%.

Gradient can be worked out with the simple equation:

Vertical Distance divided by Horizontal Distance x 100

e.g. 50 metres vertically / 100 metres horizontally
= 0.5
x 100 = 50%
or
e.g. 100 metres vertically / 50 metres horizontally
= 2
x 100 = 200%


Exercise I

What will the gradient be for the following measurements:
a) 250m vertically to 1km horizontally
b) 200m vertically to 350m vertically
c) 600m vertically to 5km vertically
d) 1300m vertically to 0.8km horizontally

Angles

Slope steepness can also be expressed as an angle in degrees (°) and some of you may find it easier to visualize a slope as an angle rather than a gradient.

Working out the angles of slope steepness can involve some reasonably complicated trigonometry, at least for those of us who are not good at maths. The quickest method for finding the angle is to simply memorize the relationship between gradients and angles.

For a complete overview of the equivalent angles and gradients, look at the graph below. It depicts a right-angled triangle of various angles.

A right-angled triangle in case you've forgotten is an triangle with an angle of 90 degrees where the horizontal line meets the vertical line (usually marked by a square). The slope is the hypotenuse of the triangle. We'll look at these later in this lesson.





If you study the graph above carefully you will notice that a gradient of 200% is not double the angle of 100% (45°) but 63°. This is because the ratio between gradient and angle decreases as the angle gets steeper so that from 0 to 20% the angle of slope increases by 6° for every 10% of gradient. Yet, from 130% to 150% it is only 2° for every 10% increase in gradient.

We recommend you memorize the relationship between gradient and slope angle especially if you are interested in route planning or leading hiking groups. However, for those who just want a more general overview of slope steepness need only memorize the following angles and gradients.

Flat (0°/0% to 9°/15%)
Easy walking.
Suitable for any grade of hiker.

Quite steep (10°/17% to 20°/36%)
Feels steep going uphill.
Fast coming downhill.
Hard on the unfit going uphill.

Steep (21°/39% to 35°/70%)
Tiring going directly up hill (probably requires zigzagging to reduce steepness). Coming directly downhill is possible but will be hard on the knees.
Less confident hikers may start to pick their way down.
Not for beginners or the unfit.

Very Steep (36°/72% to 45°/100%)
Scrambling which requires using hands for balance and leverage.
Coming downhill is slow and requires great care. Rope may be required.
For experienced hikers only.

Needless to say, anything above 45°/100% is very serious and will require climbing skills and equipment.

We'll look in more detail the how slope steepness affects different kinds of terrain but before we do we need to learn how to work out slope steepness by counting contour lines on maps.


Exercise II


Whether you use angles or gradients for assessing slope steepness is not important. But you should be able to visualize that angle or gradient. If you find out a slope is 36°/72% you should have a good idea what that looks like. The following exercises should help you do this.

Look at the right angled triangles below. Try to guess the slope steepness as an angle in degrees and as a gradient in percentages. Your estimations need to be within about 5° or 10%.



Extra Practice


At home

For this, you will need a pencil, some blank paper and a protractor.
Quickly draw several right-angled triangles of different levels of steepness. Try to guess the angle / gradient. Then check with a protractor.

Contour lines and slope steepness


The steepness of a slope can be worked out on maps by counting how many contour lines lie within a centimeter or two centimetres. Obviously, the more contour lines there are, the steeper the slope but the scale of the map must also be taken into account. For example, on a 1:25 000 map 4 thin contour lines in 1 centimeter would merely be quite steep but on a 1:6000 map 4 thin contour lines in 1 centimeter would be very very steep.

Counting contour lines






Quick estimates of slope steepness on different scaled maps


Once you have counted how many contour lines and measured the length of the slope then you need to factor in the map scale. If you don't you will make serious errors.

For example, let's say you count 4 contour lines in 1 centimetre on a 1:50 000 map.
This would be slope of about 10° or 16% - a relatively easy quite steep slope.

But, if that was 4 contour lines in 1 centimetre on a 1:10 000 map.
This would be slope of about 27° or 50% - nearly triple the steepness.

Study the three contour counts for basic slope steepnesses on different scaled maps below. The scales given are common to hiking maps especially the Hong Kong Countryside Series Maps.







Exercise III


Estimate slope steepness by counting the contour lines on the different scaled maps below. Count the contour lines from the base to the tip of the arrow.

State whether the slope is ascending or descending according to the direction the arrow is pointing in.

You may want to use the Countryside Series Maps. The map and the grid squares are given for you.

Question 1)

Countryside Series Map: Sai Kung & Clearwater Bay
Scale: 1: 25 000
Grid Squares: KK2881 to KK3183
Give the gradient and angle for the lines a) and b) and state if the slope is ascending or descending (depending on the direction of the arrow).
For example, the green line has a count of 15 thin contour lines or 3 thick contour lines in 2cm.
Therefore, the gradient or angle is 32% or 18°.
The direction of the arrow shows that it is an ascending slope.

  • Remember to consider the scale of the map!



  • Question 2)

    Countryside Series Map: Lantau & Neighbouring Islands
    Scale: 1: 25 0000
    Grid Squares: HE0063 to HE0163
    Give the gradient and angle for the lines a) and b) and state if the slope is ascending or descending (depending on the direction of the arrow).



    Question 3)

    Countryside Series Map: Hong Kong & Neighbouring Islands
    Scale: 1: 20 000
    Grid Squares: KK1563 to KK1564
    Give the gradient and angle for the lines a) and b) and state if the slope is ascending or descending (depending on the direction of the arrow).



    Question 4)

    Countryside Series Map: Hong Kong & Neighbouring Islands (Tung Lung Island)
    Scale: 1: 15 000
    Grid Squares: KK2062 to KK2162
    Give the gradient and angle for the lines a), b) and c) and state if the slope is ascending or descending (depending on the direction of the arrow).




    Effects of slope steepness



    The Foreshortening Effect

    The bad news is that as things get steeper they also get longer. To illustrate the point, let's consider an extreme example of the foreshortening effect. If you look directly down on the IFC tower in Central it only appears to be 80m wide, making it little more than a dot on a map. Yet, because it is 410 metres high, if you were to climb up and then down it, you would actually travel a distance of nearly 1 kilometre.

    This is because a map is a bird’s eye view of the landscape: we are looking directly down on the landscape. In the graphic below you can see a that on the map the slope appears to be only one kilometre long but since the slope is angle of 60° (175%), then you would actually travel twice that distance. Not only does this mean climbing more than twice the distance (2 kilometres) but this would also take double the time.




    The table below gives an outline of the extra distance travelled depending on the angle of a slope.






    Effects of slope steepness on different kinds of terrain


    Things get more complicated when we consider the fact that slopes are rarely uniform and that slopes are constantly being reshaped by man (trails and roads) and by nature (heat, cold, wind, rain and flowing water).

    Remember that navigation is often a matter of matching what you see around you to the map. On a misty day with poor visibility, locating your position may depend on interpreting the features of the slope around you.

    The following graphics illustrate what might be found on different kinds of hiking routes of different grades of steepness. These are:

    i) well-maintained distinct trails
    ii) difficult trails
    iii) stream courses
    iv) slopes with no trails

    i) Well-maintained trails (flat & steep)






    ii) Difficult trails (flat & steep)







    iii) Stream courses (flat, steep, very steep)









    iv) Slopes with no trails (flat, steep, very steep)











    Exercise IV


    Read the section on the effects of slope steepness above and then answer the questions in the quiz below.

    1) On which kinds of trails and on what angle/gradient of slope steepness are you likely to find rest shelters?
    2) Can you trust ropes left by hikers? Why?
    3) If you are lost and you see climbing anchors, why is that a good sign?
    4) On what angle/gradient of slope are you likely to find scree? What are the dangers of climbing on scree?
    5) If you travel 2km on a slope with a steepness of 40° / 83%, how far will you actually travel?
    6) How can you tell if the slope on a map is ascending or descending?
    7) On what things might you find tags and chalk arrows left by hikers to show the way.
    8) At what angle / gradient are you likely to find high impassable waterfalls on a stream course?
    9) Name three obstacles you may have to overcome when hiking along flat ground with no trails.


    Extra Practice


    At home
    Find a road near your home which has the gradient given on a road sign. Hike up and down it to get a feel for that gradient.

    Hiking Practice
    The next time you are hiking pay particular attention to the features you find on the various slopes you encounter. Check your map and count the amount of contour lines in the area you are hiking in.

    Calculating slope steepness and slope length


    The following section is for hiking leaders who want to attempt difficult off-trail routes in areas they have not hiked in before.

    The calculations should be done in preparation for the hike. Obviously, making calculations in difficult hiking situations is leaving it too late.

    Slope steepness


    To calculate slope steepness on a given map follow these steps:

    For example, let's say you counted 3 thin contour lines in 1 centimetre on a 1:20 000 map.

    Step 1
    Count the amount of contour lines on the slope and convert that to vertical distance in metres.
    e.g. 3 x 20 metres = 60 metres

    Step 2
    Multiply the length you measure on the map by the map scale to get the horizontal distance in metres.
    e.g. 1 centimetre x 20 000 = 20 000 centimetres (i.e. 200 metres)

    Step 3
    Divide the vertical distance by the horizontal distance and multiply it by 100 to get the gradient.
    e.g. 60 / 200 = 0.3 x 100
    = 30%

    Step 4
    Go to the graphic comparing gradients and angles near the beginning of this lesson and figure out the nearest approximate angle.
    e.g. 30% is 17°

    Tip
    If you are having trouble visualizing an angle then draw it on a piece of paper using a proctractor.

    Finding angles for yourself
    If you are good at maths you can use trigonometry to find the angle. But remember this kind of accuracy isn't needed for a hiker!

    Another solution to find the angle for a given gradient is to draw a triangle rather like the one earlier in the lesson with your horizontal distance as the base of the triangle and the vertical distance making the right angle. This has the added advantage of being able to see the angle too.

    In our example, we can convert our measurements to centimetres of the same ratio:
    e.g. Horizontal distance = 200 metres Vertical Distance = 60 metres
    200 to 60
    20cm to 6cm (Hmm. This makes a rather large triangle so let's half the numbers.)
    10cm to 3cm
    We now have a right-angled triangle with a base of 10cm (x-axis) and a vertical height of 3cm (y-axis).
    Now all you need to do is draw the slope and measure the angle of the triangle (at the tip of its base) with a protractor.

    Exercise V


    For this exercise, you will need: a pencil, a rubber, a ruler in centimetres, some blank paper (A4), a protractor, a calculator
    Remember, this exercise is based on contour lines for the Hong Kong Countryside Series Map where thin contour lines equal 20 metres of vertical distance and thick contour lines equal 100 metres of vertical distance.

    Work out the angle of the slope for these measurements.

    1) You count 4 thin contour lines over 2 centimetres on a 1:15 000 map.

    2) You count 2 thick contour lines followed by 3 thin contour lines over 1½ centimetres on 1:25 000 map.

    3) You count 2 thin contour lines then 1 thick contour line then 4 thin contour lines then 1 thick contour line then 1 thin contour line over 4 centimetres on a 1:6 000 map. You may want to quickly sketch these contour lines along a 4cm line to help you solve this problem.

    4) You count 4 thin contour lines over ½ (0.5) centimetre on 1:10 000map.

    Extra Practice

    At home
    Look at a hilly area on your map. Find the steepest slope in a grid square or within a country park.

    Hiking
    i) Ask your hiking leader for the next route. Find the steepest slope on the hike.
    ii) Next time you are hiking, see if you can tell where you are only by considering slope steepness and its effects.

    Effects of slope steepness in more detail


    Previously, we looked at the effects of slope steepness quite generally. The tables below give a more detailed description of each level of steepness. This part is really for hiking leaders when planning hikes.












    Slope length


    For this section and exercise, you will need: a pencil, a rubber and a calculator with a square root function (marked “√”).

    If we take the horizontal and vertical distance as the sides of a right-angled triangle, we can calculate the length of the slope because the slope length is the hypotenuse of the right-angled triangle.



    Remember your trigonometry lessons at school?
    ”The length of the hypotenuse is the square root of the sum of the squares of the lengths of the other two sides.” Okay. Maybe not so follow the steps below.

    Step 1
    Square the horizontal and vertical distance
    Horizontal Distance = 400 metres
    Squared: 400x400 = 160000

    Vertial Distance = 300 metres
    Squared 300x300 = 90000

    Step 2
    Add them together
    160,000 + 90,000 = 250,000

    Step 3
    Take the square root of that number
    √250,000 = 500
    Slope Length = 500 metres

    Exercise VI


    Calculate the length of the following slopes.

    a)
    Horizontal Distance = 300m
    Vertical Distance = 450m

    b)
    Horizontal Distance = 1200m
    Vertical Distance = 240m

    c)
    Horizontal Distance = 200m
    Vertical Distance = 420m

    d)
    Horizontal Distance = 3.1km
    Vertical Distance = 865m

    e)
    You count 7 thin contour lines in 2.5 centimetres on a 1:15 000 map.












    Answers


    Exercise I


    a) 25% b) 57% c) 12% d) 162.5%

    Exercise II


    1) 60° (175%)
    2) 15° (21%)
    3) 30° (59%)
    4) 24° (47%)

    Exercise III

    Questions 1
    a) 4 thin contour lines per 1cm = 32% / 18° (Steep); descending
    b) 8 thin contour lines per 2cm (i.e. 4 thin contour lines per 1cm) = 32% / 18° (Steep); ascending

    Questions 2
    a) 9 thin contour lines per 1cm = 72% / 36° (Very Steep); ascending
    b) 0 thin contour lines per 1cm; 0% / o° (flat)

    Questions 3
    a) 2 thin contour lines per 1cm = 20% / 12° (Quite Steep); ascending
    b) 4 thin contour lines per 4cm (i.e. 1 thin contour lines per 1cm) = 10% / 6° (Almost flat); descending

    Questions 4
    a) 7 thin contour lines per 1cm = 93% / 43° (Very Steep); ascending
    b) 4 thin contour lines per 2cm (i.e. 4 thin contour lines per 1cm) = 53% / 28° (Steep); ascending
    c) 1 to 2 thin contour lines per 1cm = 13% / 7° (Quite steep); ascending, descending, ascending

    Exercise IV

    1) Steep (25° / 47%), named / well-maintained / major trails and footpaths
    2) No. The rope may be old and weak or the anchor might be a dying tree or a loose rock. Test thoroughly before committing to it or use it only for balance, not for leverage.
    3) Climbing areas are usually connected to good trails or at least have a well-trodden route out and in.
    4) Usually around 40° / 83% and above. Scree in Hong Kong is often covered by vines and creepers making it easy to stumble and sprain ankles or worse.
    5) 1km on a slope of 40° / 83% means about 1310m of actual travel. Therefore 2km would be 2 x 1310m = 2620m or 2km.
    6) Check the numbers written on the thick contour lines.
    7) Tags are usually found hanging from trees or other plants. Chalk arrows can be found on boulders or rocks.
    8) 45° (100%) However, they can occur on angles from 30° (59%)
    9) Swamps, thick vegetation, large boulders, ditches


    Exercise V

    1) You count 4 thin contour lines over 2 centimetres on a 1:15 000 map.
    Vertical distance: 4 x 20 = 80 metres
    Horizontal distance: 2 x 150 = 150 metres
    80 / 150 x 100
    = 53%
    Approximately 28°

    2) You count 2 thick contour lines followed by 3 thin contour lines over 1½ centimetres on 1:25 000 map.
    Vertical distance: 2 x 100 = 200 metres + 2 x 20 metres = 40 metres. Total: 240 metres
    Horizontal distance: 1.5 x 250 = 375 metres
    240 / 375 x 100
    = 64%
    Approximately 33°

    3) You count 2 thin contour lines then 1 thick contour line then 4 thin contour lines then 1 thick contour line then 1 thin contour line over 4 centimetres on a 1:6 000 map. You may want to draw these contour lines to help you solve this problem.
    Vertical distance: 9 x 20 = 180 metres (Remember you count thick contour lines as 20 metres when they are in a sequence of thin contour lines. If you add up all the contour lines it comes to 9.)
    Horizontal distance: 4 x 60 = 240 metres
    180 / 240 x 100
    = 75%
    Approximately 37°

    4) You count 4 thin contour lines over ½ (0.5) centimetre on 1:10 000map.
    Vertical distance: 4 x 20 = 80 metres
    Horizontal distance: 0.5 x 100 = 50 metres
    80 / 50 x 100
    = 160%
    Approximately 59°
    (You could draw a triangle with a base (x-axis) of 2.5cm and 4cm tall (y-axis) and measure the angle with a protractor to get this figure.)

    Exercise VI


    a)
    Horizontal Distance = 300m
    Vertical Distance = 450m
    300 x 300 + 450 x 450
    90,000 + 202,500 = 292,500
    √292,500 = approx. 1224
    Slope length = 1224m

    b)
    Horizontal Distance = 1200m
    Vertical Distance = 240m
    1200 x 1200 + 240 x 240
    1,440,000 + 57,600 = 1,497,600
    √1,497,600 = approx. 1224
    Slope length = 1224m

    c)
    Horizontal Distance = 200m
    Vertical Distance = 420m
    200 x 200 + 420 x 420
    40,000 + 176,400 = 216,400
    √216,400 = approx. 465
    Slope length = 465m

    d)
    Horizontal Distance = 3.1km
    Vertical Distance = 865m
    3100 x 3100 + 865 x 865
    9,610,000 + 748,225 = 10,358,225
    √10,358,225 = approx. 3218
    Slope length = 3218m


    e)
    You count 7 thin contour lines in 2.5 centimetres on a 1:15 000 map.

    Horizontal Distance = 2.5 x 150 = 375 metres
    Vertical Distance = 7 x 20 = 140 metres
    375 x 375 + 140 x 140 = 160225
    √160225 = 400 (approx.)
    = 400 metres (approx.)

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