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GEOMETRYCoordinate GeometryMathematics Calculator

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Rise Over Run

Slope = rise/run = (y₂−y₁)/(x₂−x₁). Rise is vertical change; run is horizontal. Percentage grade = slope × 100%. Used for road grades, roof pitch (e.g. 4:12), wheelchair ramps (max 1:12), and stairs.

Concept Fundamentals

y₂ − y₁

Rise

x₂ − x₁

Run

m = rise/run

Slope

m × 100%

Grade

Find Rise, Run, SlopeEnter two points

Why This Mathematical Concept Matters

Why: Rise over run is the foundation of slope. Road grades (6% = 6 ft per 100 ft), roof pitch (4:12), ADA ramps (1:12 max), and ski slopes all use this measure.

How: Rise = y₂ − y₁, Run = x₂ − x₁. Slope m = rise/run. Angle θ = arctan(m). Percentage grade = m × 100. Vertical line: run=0, slope undefined.

●6% road grade = 6 ft rise per 100 ft run.
●4:12 roof pitch = 4" rise per 12" run.
●ADA ramp max 1:12 (8.33% grade).

Sources:Khan Academy — SlopeWolfram MathWorld — Slope

Sample Examples — Click to Load & Calculate

Enter Two Points

Point 1

x₁

y₁

Point 2

x₂

y₂

⚠️For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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Slope = rise/run = Δy/Δx.

— Coordinate Geometry

Grade = slope × 100%.

— Application

Key Takeaways

• Rise = $y_2 - y_1$ (vertical change)
• Run = $x_2 - x_1$ (horizontal change)
• Slope = rise/run = $\\frac{y_2 - y_1}{x_2 - x_1}$
• Angle = $\\arctan(m)$ (with x-axis)
• Percentage grade = slope × 100% (e.g. road grades, roof pitch)

Did You Know?

Road Grades

A 6% grade means the road rises 6 feet for every 100 feet horizontally. Steep roads can be 10–15%.

Roof Pitch

Roof pitch is often given as rise:run (e.g. 4:12 means 4" rise per 12" run).

Wheelchair Ramps

ADA recommends max 1:12 slope (8.33%). A 1" rise requires 12" of run.

Ski Slopes

Green runs ~15–25%, blue ~25–40%, black 40%+. Vertical drop / horizontal distance.

Stairs

Typical step: 7" rise, 11" run. Rule of thumb: 2×rise + run ≈ 25 inches.

Positive vs Negative

Positive slope: line rises left to right. Negative: line falls. Zero: horizontal.

Understanding Rise Over Run

Given two points $(x_1, y_1)$ and $(x_2, y_2)$ :

m = \\frac{\\text{rise}}{\\text{run}} = \\frac{y_2 - y_1}{x_2 - x_1}

The slope m is the rate of change of y with respect to x. Percentage grade = m × 100.

Expert Tips

Order Doesn't Matter

Swapping P₁ and P₂ negates both rise and run, so the slope stays the same.

Vertical Line

When run=0 (same x-coordinate), slope is undefined. The line is vertical.

Horizontal Line

When rise=0 (same y-coordinate), slope is 0. The line is horizontal.

Grade Interpretation

10% grade = 10 units up per 100 units forward. Common for steep driveways.

Frequently Asked Questions

What is rise over run?

Rise is the vertical change (Δy); run is the horizontal change (Δx). Slope = rise/run.

What is percentage grade?

Grade = slope × 100%. A 5% grade means 5 units up per 100 units forward.

When is slope undefined?

When the two points have the same x-coordinate (run=0). The line is vertical.

What is a 1:12 slope?

Rise 1, run 12. Slope = 1/12 ≈ 0.083. Common for wheelchair ramps.

How do I read roof pitch 4:12?

4 inches of rise for every 12 inches of run. Slope = 4/12 = 1/3.

What does negative slope mean?

The line falls from left to right. Rise is negative (y decreases as x increases).

How is this used in construction?

Road grades, roof pitch, ramp design, drainage slope, and stair design.

How to Use This Calculator

Enter the coordinates of two points, or click a sample example.
Click Calculate to get rise, run, slope, angle, and percentage grade.
Review the visualization showing rise and run.
Check the step-by-step solution and copy results if needed.

Disclaimer: This calculator is for educational use. For construction, verify with local codes (e.g. ADA ramp requirements).

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