Triangle Slope Calculator - Rise, Run, Slope, and Angles

A triangle slope calculator finds the slope of a right triangle from its rise and run and turns that one ratio into the slope as a decimal, the slope as a percent, the two acute angles of the triangle, and the hypotenuse. The same geometry shows up in algebra, trigonometry, physics, and construction, so one tool that returns every form of the answer covers most problems.

• • Algebra and coordinate geometry homework: Find the slope of a line drawn next to a right triangle on a coordinate grid and read off the angle it makes with the run.
• • Physics and engineering graphs: Convert the rise and run of a graphed segment into a slope, an angle, a percent grade, and a hypotenuse length for vectors or rate plots.
• • Construction and grade checks: Translate two elevation points into a slope percent and an angle for ramps, driveways, drainage, and roof pitches.
• • Data analysis quick check: Compute the slope of a best-fit line from two anchor points and see the same slope as the tangent of alpha in one pass.

The output is one slope plus the same value written as a percent, the two acute angles, and the hypotenuse, all derived from the same rise-and-run input.

When the two coordinates are given instead of a rise and a run, the Slope Calculator reads (x1, y1) and (x2, y2) and returns the same rise-over-run ratio plus the line equation in y = mx + b form.

The slope of a right triangle is the length of the vertical leg divided by the length of the horizontal leg, with a sign that depends on whether the line is rising or decreasing. The page takes your rise and run, applies that ratio, and derives the percent, the two acute angles, and the hypotenuse from the same input.

• rise (a): Vertical leg of the right triangle (the rise).
• run (b): Horizontal leg of the right triangle (the run). Must be greater than 0.
• s: Sign of the slope: +1 for a rising line, -1 for a decreasing line.
• alpha: Acute angle between the run and the hypotenuse, equal to arctan(rise / run).
• beta: Acute angle between the rise and the hypotenuse, equal to 90 - alpha.
• c: Hypotenuse of the slope triangle, equal to sqrt(rise^2 + run^2).

The percent output is the decimal slope times 100, alpha is the arctangent of the rise-over-run ratio, and beta is the complementary acute angle so that alpha + beta = 90. The hypotenuse comes from the Pythagorean theorem on the same two legs.

The same formula covers negative slopes and the horizontal case: when the line is decreasing the sign flips so the slope and the percent are negative while the angles and the hypotenuse stay positive, and when the rise is 0 the slope is 0 with alpha equal to 0 and beta equal to 90.

According to Omni Calculator, the slope of a right triangle is the rise divided by the run, the same ratio is the tangent of the angle opposite the rise, and the reciprocal of the tangent of the other acute angle gives the same slope.

The hypotenuse on this page is the same c = sqrt(a^2 + b^2) result the Pythagoras Triangle Calculator returns from the two legs, so the slope triangle closes the loop between the slope, the two angles, and the segment length.

A slope triangle is a right triangle drawn next to a line so that the rise and run become two of its legs. These four concepts cover the cases you will run into most often.

Alpha and beta are the two acute angles in the slope triangle, so alpha + beta = 90 for any positive run, and the same ratio is read three different ways: as a slope, as a tangent, and as a cotangent.

Because slope equals tan(alpha), the Arctan Calculator is the natural next step when you already know the slope and want alpha in degrees, or when you know alpha and want the slope back out.

Enter the rise and the run of the right triangle, then pick whether the line is rising or decreasing. The result panel updates as soon as any field changes, so the slope, percent, angles, and hypotenuse refresh in real time.

1. 1 Enter the rise: Type the vertical leg (a) of the slope triangle. Use 0 for a horizontal line or a positive number for any other case.
2. 2 Enter the run: Type the horizontal leg (b) of the slope triangle. The run must be greater than 0, because the formula divides by it.
3. 3 Pick the line direction: Choose 'rising' for a positive slope or 'decreasing' for a negative slope. The hypotenuse and the angles are always positive.
4. 4 Read the slope and percent: The result panel shows the slope as a decimal and as a percent grade, signed based on the direction you chose.
5. 5 Read the two angles and the hypotenuse: Use alpha to see the angle the line makes with the run, beta for the complementary angle, and the hypotenuse for the length of the sloped segment.
6. 6 Fix a zero run: If the run is 0, the calculator refuses to compute and asks for a positive run, because dividing by 0 is not defined.

A wheelchair ramp rises 0.6 m over a run of 6 m. Enter rise 0.6, run 6, and pick 'rising'. The result panel shows slope 0.1, percent 10%, alpha 5.7106 degrees, beta 84.2894 degrees, and hypotenuse about 6.0299 m, which is the slope and angle of the ramp in one pass.

When the only number you need is the percent grade for a ramp, driveway, or roof, the Slope Percentage Calculator reads a rise and a run directly and skips the angles and the hypotenuse.

The result tells you exactly which number is the slope and what it means in percent, angle, and triangle form, all in a single pass.

• • Direct rise-and-run calculation: Returns the slope, percent, both acute angles, and the hypotenuse from a rise and a run in one pass.
• • Signed slope with a direction picker: Uses a sign based on the line direction, so the same rise and run produce a positive or negative slope without extra steps.
• • Tangent and Pythagorean outputs: Builds alpha, beta, and the hypotenuse from the same input, so the slope, the angle, and the segment length stay in sync.
• • Zero-run and horizontal-line handling: Rejects a zero run with a clear validation error and treats a zero rise as a horizontal line with a slope of 0 and a 90 degree beta.
• • Real-time recalculation: Updates the result panel as soon as the rise, run, or direction field changes.

Rerun the calculation whenever a new pair of legs is given; the formula is the same signed rise-over-run division regardless of sign.

A few characteristics of the input pair change the readability of the result, even when the underlying math is the same.

• • The calculator handles a single slope triangle at a time. For batch work, copy the (rise, run) pairs into a spreadsheet and apply the formula to each row.
• • The result is rounded for display only. The internal value uses the full double-precision division, so chaining the slope with a follow-up calculation should use the unrounded rise-over-run result.
• • The calculator does not handle complex legs. The rise and run must be real numbers, and the run must be greater than 0.

When the input pair comes from elevation points, the slope percent matches the percent grade used in accessibility ramps and drainage, and the angle alpha matches the angle a roof pitch calculator converts to in one step.

According to Khan Academy, the tangent of an acute angle in a right triangle is the length of the opposite leg divided by the length of the adjacent leg, and the two acute angles in a right triangle add up to 90 degrees.

According to Wolfram MathWorld, the slope of a straight line is the tangent of the angle the line makes with the positive x-axis, and for two points (x1, y1) and (x2, y2) on the line the slope equals (y2 - y1) divided by (x2 - x1).

On a real construction project, the same rise-over-run ratio drives the roof pitch in inches per foot (multiply the slope percent by 0.12 to get inches of rise per foot of run), and the Roof Pitch Calculator converts that pitch to degrees and to a pitch ratio in one step.