Triangle Slope Calculator

Enter two points to find the slope, angle of inclination, and hypotenuse length of the right triangle (slope triangle) formed between them, plus the line's equation.

Quick Facts

Slope formula
m = Δy / Δx = (y₂-y₁)/(x₂-x₁)
Rise over run between the two points.
Angle of inclination
θ = arctan(m)
Angle the line makes with the horizontal.
Hypotenuse (Pythagorean theorem)
h = √(Δx² + Δy²)
Straight-line distance between the two points.

Your Results

Calculated
Slope (m)
-
m = Δy ÷ Δx (rise over run)
Angle of Inclination
-
θ = arctan(m), from horizontal
Hypotenuse Length
-
h = √(Δx² + Δy²)
Line Equation
-
Slope-intercept form: y = mx + b

Ready

Enter two points, then press Calculate.

Formula and Method for Triangle Slope

A "slope triangle" is the right triangle formed underneath a line segment when you drop a vertical leg (the rise) and a horizontal leg (the run) between two points. Its legs let you compute the line's slope directly from coordinates: m = Δy / Δx = (y₂ - y₁) / (x₂ - x₁). This calculator takes two points, builds that right triangle, and reports the slope, the angle the line makes with the horizontal, the length of the hypotenuse (the straight-line distance between the points), and the line's slope-intercept equation.

How the calculation works

Enter the coordinates of two points. The calculator finds the horizontal leg (run = x₂ - x₁) and vertical leg (rise = y₂ - y₁) of the slope triangle, then divides rise by run to get the slope m. The angle of inclination comes from θ = arctan(m), giving the angle in degrees between the line and the positive x-axis. The Pythagorean theorem, h = √(run² + rise²), gives the hypotenuse — the straight-line distance between the two points. Finally, the y-intercept b = y₁ - m·x₁ completes the equation y = mx + b.

Common mistakes

  • Reversing the order of subtraction: rise and run must use the same point order — (y₂-y₁) over (x₂-x₁), not (y₁-y₂) over (x₂-x₁). Swapping only one flips the sign of the slope.
  • Vertical lines: if the two points share the same x-coordinate, run = 0 and the slope is undefined (a vertical line has no finite slope).
  • Confusing slope with angle: a slope of 1 means a 45° angle, but a slope of 2 is about 63.4°, not 90° — the relationship is arctan, not linear.

Real-world applications

  • Road grades, ramps, and roof pitches are expressed as slope (rise over run) or as a percentage grade.
  • Physics and engineering use the angle of inclination to resolve forces along an incline.
  • Coordinate geometry uses slope to test whether lines are parallel (equal slopes) or perpendicular (negative reciprocal slopes).
  • Surveying and construction use the hypotenuse length to lay out diagonal braces, stringers, or sightlines between two known points.

Frequently Asked Questions

What is the formula for the slope of a triangle's hypotenuse?
Slope is rise over run: m = (y₂ - y₁) / (x₂ - x₁), where the two points are the endpoints of the line and the rise and run are the vertical and horizontal legs of the right triangle formed beneath it.
How do I find the angle of inclination from the slope?
Take the arctangent of the slope: θ = arctan(m). For example, a slope of 1 gives θ = arctan(1) = 45°.
Why is the slope undefined for a vertical line?
A vertical line has zero horizontal run (x₂ = x₁), and dividing rise by zero is undefined. Vertical lines are described by x = constant instead of y = mx + b.
How is the hypotenuse of the slope triangle related to the distance formula?
The hypotenuse is exactly the distance between the two points: h = √((x₂-x₁)² + (y₂-y₁)²), which is the Pythagorean theorem applied to the horizontal and vertical legs.

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