How this instrument works
The slope of a line is the ratio of vertical change to horizontal change between any two of its points: m = (y₂ − y₁) ⁄ (x₂ − x₁), rise over run. A value of 2 means the line gains two units of height for every unit it travels right; −0.5 means it drops half a unit over the same distance. Zero marks a horizontal line, and the sign alone tells you at a glance whether the line rises or falls as you read it left to right.
Because tangent is the ratio of opposite to adjacent in a right triangle, the slope is exactly the tangent of the line's angle of inclination. Running the arctangent backwards recovers that angle: θ = atan(m), expressed here in degrees. The relationship is deliberately nonlinear — m = 1 is 45°, but m = 2 is only 63.43°, and m = 10 is 84.29°. Steepness in ratio terms outruns steepness in angle terms, which is why the angle crowds toward 90° yet never arrives.
One case escapes the formula: when x₂ = x₁ the run is zero, the division is undefined, and the line is vertical. That is not a flaw in the arithmetic but a fact about the definition — a vertical line has no rise per unit of run because it never runs. Engineers dodge the singularity by quoting grade as a percentage (m × 100) for roads and ramps, a convention this sheet's result converts to with one mental shift of the decimal point.
- Enter the first point's coordinates in the Point A fields, x₁ and y₁.
- Enter the second point in the Point B fields, x₂ and y₂ — decimals and negatives are both welcome.
- Read Slope (m): the vertical change per unit of horizontal travel between your two points.
- Read Angle (°): the inclination from horizontal, positive for a rising line, negative for a falling one.
- If the sheet flags x₂ = x₁, your line is vertical — no finite slope exists, and the angle would be ±90°.
Worked example — from (1, 2) to (5, 10)
Take the points (1, 2) and (5, 10). Rise: 10 − 2 = 8. Run: 5 − 1 = 4. Slope: m = 8 ⁄ 4 = 2 — the line climbs two units for every unit it moves to the right, steeper than any staircase you would want to use.
The angle follows from the arctangent: θ = atan(2) = 63.43°. Notice it is not double the 45° of a slope-1 line — the angle grows ever more slowly as m increases, crowding toward 90° but never reaching it.
Questions
What does a negative slope mean?
The line falls as you move left to right: y decreases while x increases. The steepness reading is symmetric — a slope of −2 descends exactly as steeply as +2 climbs — and the angle output mirrors it, showing −63.43° instead of +63.43°. Sign is direction; magnitude is steepness.
Why is the slope of a vertical line undefined?
Because the run is zero and division by zero has no value. Between two points with the same x-coordinate, the line gains height without moving horizontally, so rise per unit of run asks a question with no answer. It is not infinity in the arithmetic sense — the ratio simply does not exist, which is why this sheet raises a flag rather than printing a number.
How do I turn slope into a percent grade?
Multiply by 100. A slope of 0.08 is an 8% grade — the figure on road signs — meaning 8 units of climb per 100 of horizontal travel. Note that a 100% grade is m = 1, which is only 45°, not vertical; highway engineering rarely exceeds 6–7% on main routes, and even the famously steep San Francisco streets top out near 31.5%.
Does it matter which point I call Point A?
No. Swapping the points negates both the rise and the run, and the two sign changes cancel: (y₁ − y₂) ⁄ (x₁ − x₂) equals (y₂ − y₁) ⁄ (x₂ − x₁). Keep each point's coordinates paired correctly, though — mixing one point's x with the other point's y is the error that actually bites.
Can the angle output ever reach 90 degrees?
No. The arctangent approaches ±90° as m grows without bound but never gets there — m = 10 reads 84.29°, m = 100 reads 89.43°, m = 1000 reads 89.94°. A true 90° line is vertical, has no finite slope, and is exactly the case the x₂ = x₁ check intercepts before the division is attempted.