Triangles & geometry in building framing:
This article describes how we can use a simple tape measure and basic aritematic in framing a building deck, floor, wall or roof to get it square and straight. How to Use the 6-8-10 Rule to Square Up any Deck, Wall, Floor, Roof. How to Calculate stair tread depth or riser height from stairway slope in degrees
We also discuss stair step rise & run calculations when we are given only the slope of a stairway, and we explain how to use the tangent function to convert a measured angle into stair step rise & run dimensions.
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We can and often do use this basic right triangle function a2 =b2 +c2 in deck building to be sure that we are placing the deck sides at right angles to the building by using what my carpentry teacher (Bernie Campbalik) called the 6-8-10 rule.
The 6 8 10 framing square-up rule works because (62 + 82) = 102 or 36 + 64 = 100. But that's enough plane geometry for now.
For a different and interesting use of triangles and plane geometry to convert stair slope in degrees to tread depth and riser height see Calculate stair tread depth or riser height from stairway slope in degrees.
The geometry approach to calculating stair rise and run over rough terrain when you have only two measurements works correctly provided you know enough triangular geometry to convert between your sloped-line measurement (the hypotenuse of the triangle) and the triangle's other sides that form the rise & run dimensions. Using a calculator and it's square root function, and the most basic knowledge of geometry is not enough - we also need a table of sine and cosine values, functions readily found these days online. 
a2 =b2 +c2 - the square of the length of the hypotenuse (a) equals the squares of the lengths of the opposite sides of a right triangle (b) and (c).
In a geometry or trigonometry text you'll see that when we refer to the angle ab we mean the angle formed at the intersection of lines a and b, or in this case the lower left angle in our sketch.
And so the three angles formed by the triangle's sides are ab, bc, and ac.
Knowing that the triangle abc will always include a 90 degree right angle ( i.e. this is always a "right triangle"- angle bc in our sketch) allows use of sine and cosine tables to obtain the lengths of the two unknown sides b and c in the sketch.
Geometry teaches that if we know three pieces of data about a right triangle (one side length and two angles, or two side lengths and one angle) we can obtain all of the other data about a triangle - all of its angles, and the actual lengths of all of its sides.
sin of angle ab = (length of the opposite side c / the triangle's hypotenuse a) = a / c
cos of angle ab = (length of the adjacent side b / the triangle's hypotenuse a) = b / a
But the rub is that we need to know at least two of the angles on this triangle. We know bc is always 90 degrees. On the ground we'd have to measure either angle ab or angle ac. You can do this using a transit, a protractor or other angle measuring tool either by placing your angular measuring tool on the sloped surface of the stringer side a, or by actually measuring the angle formed between side a and a horizontal or level surface.
Our carpenter's square at left is one such tool, though there are some neat level tools that, placed on a slope, will simply give you the angle directly as a readout.
Example: if our stairs are to run from the two extreme ends of the 2x stringer we illustrate at above right, and if that length (after cutting to "fit" the hill and desired stair run) measured 100 inches exactly, using just that known length of the hypotenuse of the triangle we would obtain the key measurements of stair rise c and stairway run b by first converting all measurements to inches and then using a table of sine and cosine values. Suppose we measure angle ab (sketch above) at 30 degrees.
sin of angle ab = (length of the opposite side c / the triangle's hypotenuse a) = a / c = 100 / c
cos of angle ab = (length of the adjacent side b / the triangle's hypotenuse a) = b / a = b / 100
Reading from sine and cosine tables, we'll find that side c (rise) = 50" and that side b (run) = 86.6"
If you don't think this whole sine-cosine process is absolutely horrible for field use in building a set of stairs then we're not on the same planet.
Stairway at 38 degrees: what is the rise and foot? - George Tubb
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George we're back to plane geometry. It's not really that bad if we just realize this is simple plane geometry that defines the relationships between a right triangle (that means one angle of the triangle is set at 90 degrees) and the lengths of its sides.
A generalized example of the explanation below is given at ROOF SLOPE DEFINITIONS where we explain that you can easily convert any slope (roof, road, or stair) into rise or run distances using the Tangent function.
In geometry, if we know the lengths of sides of a triangle, we can calculate its angles. If we know two of its angles we can calculate the lengths of its sides. Etc.
The slope of your stairs is given as 38 degrees. And we figure that in calculating (or measuring) the "rise" of a stairway we can assume we are not so stupid as to not hold our tape vertical between floors - so we can assume the other known angle is 90 degrees - we've got a nice "right triangle".
You are asking the rise and "run" of the treads. There is no single answer, since we could choose different tread depths or "runs" that would give different tread rises or heights. E.g. we could make the stairs "one giant step" or "three little steps".
But to stay within reason, we can pick a desired step run or depth or step height or rise, and calculate the second number with the help of a calculator that will convert an angle in degrees using the Tan (tangent) function. \
Choosing a stair tread depth (step run length) of 10" in the equation above gave us a calculated individual step rise of 7.8". That is, on a 38 degree sloped stairway, we will ascend 7.8" in height for every 10 inches of horizontal distance traveled.
If we chose instead to use a stair tread individual rise of 7.0" we would calculate as follows:
0.7813 = 7.0" / X1
X1 x (0.7813) = 7.0"
X1 = 7.0" / (0.7813)
X1 = 8.9" step run or tread depth.
This is not a good tread depth (too narrow) which reflects that basically, a 38 degree slope stairway is too step for comfortable climbing and forces us to have either treads that are too shallow or risers that are too high.
A tangent is the ratio of two sides of a right triangle: specifically the height (Y) divided by the base or length (X). For any given stair slope (or angle) or triangle slope (angle T or "Theta" as we say in geometry class), that ratio remains unchanged.
Or in geometry speak:
Height Y1 / Length X1 = Height Y2 / Length X2
as long as we keep the slope or angle unchanged.
The tangent function is a ratio of horizontal run X and vertical rise Y. For any stairway of a given angle or slope (say 38 degrees in your case) the ratio of run (x) to rise (y) will remain the same.
[Click to enlarge any image]
That's why once you set your stair slope (too steeply) at 38 degrees, we can calculate the rise or run for any stair tread dimension (tread depth or run or tread height or riser) given the other dimension (tread height or rise or tread depth or run).
The magic of using the Tangent function is that we can use that ratio to convert stair slope or angle in degrees to a number that lets us calculate the rise and depth or run of individual stair treads
Here are two examples of stair tread run (depth) and riser height for a stair with a 38 degree slope: :
We represent this giant 23.4" tall step in our sketch at left. You can see that the triangle and angle are unchanged.
The magic is that the ratio of the rise over run (Y/X) for both sets of stairs would always be the same - because they are built to the same slope or angle. You can see that reflected in our drawing above.
For a special use of right triangles to square up building framing, also see Use the 6-8-10 Rule - a simple method for assuring that framing members have been set at right angles to one another.
Could we calculate the tangent of 38 degrees? Well it's easier to use a scientific calculator and just ask for the Tangent of a known angle. If we knew that we had a triangle of 38 degrees at angle T (Theta) and if we knew two specific measurements X and Y we could indeed calculate T = Y/X. After all, the tangent of angle Theta is the ratio of Y/X.
I used an online calculator available at http://www.creativearts.com/scientificcalculator/ and the simple formula shown in my illustration.
We chose (to make the math easy) a run of 10 inches - that's your tread depth shown as the blue X at the left end of the sketch. The tread rise (for which we are solving the equation) is the blue Y at the left of the sketch. Just to make clear that we're simply working with a "right triangle" I show X and Y again as the horizontal and vertical sides of the triangle respectively.
I got also some help (a refresher on geometry) from Ferris High school's excellent geometry department who provides a more detailed analysis of the same problem as that posed by George Tubb's question.
Those Ferris High kids in Spokane can also show you how to work this problem in the other direction: that is, if we know the rise and run of the stair we can calculate its slope or angle in degrees by using the arctangent function. -DF
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