Slope & Distance Calculator
Calculate Slope & Distance Between Two Points
Enter coordinates of two points to find slope and distance
Calculate Grade Percentage
Enter rise and run to calculate grade percentage
Calculate Angle from Slope
Convert slope to angle in degrees or radians
Find Point Given Slope & Distance
Find second point given starting point, slope, and distance
Slope & Distance Results
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Slope (m) = rise/run = (y₂-y₁)/(x₂-x₁). Distance = √((x₂-x₁)² + (y₂-y₁)²). Grade % = (rise/run) × 100. Angle = arctan(slope) in radians or degrees.
Complete Slope & Distance Guide
Slope, distance, and angle are fundamental concepts in geometry, surveying, and construction. This guide explains how to calculate these values and their real-world applications.
Key Formulas
| Concept | Formula | Description |
|---|---|---|
| Slope (m) | m = (y₂ - y₁) / (x₂ - x₁) | Rise divided by run. Rise = vertical, Run = horizontal |
| Distance | d = √((x₂-x₁)² + (y₂-y₁)²) | Pythagorean theorem. Distance between two points |
| Grade % | Grade = (rise/run) × 100 | Slope expressed as percentage. Common in roads/slopes |
| Angle (radians) | θ = arctan(m) | Angle in radians from slope value |
| Angle (degrees) | θ = arctan(m) × (180/π) | Angle in degrees. Most common format |
Slope Values & Interpretations
| Slope Value | Grade % | Angle | Interpretation |
|---|---|---|---|
| 0 | 0% | 0° | Horizontal line (flat) |
| 0.5 | 50% | 26.57° | Moderate uphill slope |
| 1.0 | 100% | 45° | 45° angle (equal rise and run) |
| 2.0 | 200% | 63.43° | Steep uphill slope |
| ∞ (undefined) | ∞ | 90° | Vertical line (straight up) |
| -1.0 | -100% | -45° | Downhill slope at 45° |
Real-World Applications
- Road & Ramp Gradients: Maximum safe slope for vehicles. Wheelchair ramps: max 1:12 (8.3%). Steep hills: 10-15% grade
- Roofing: Roof pitch (e.g., 4:12 means 4" rise per 12" run). Steeper pitch sheds water better
- Stairs: Building code requires slope 30-35° (ratio ~7:10). Too steep = unsafe, too shallow = odd walking
- Surveying & Mapping: Determining terrain slope. Slope stability in hillsides
- Navigation: Finding bearing between two GPS coordinates using slope/angle
- Construction & Grading: Drainage requirements. Slope for water runoff typically 1-2%
- Sports: Ski slope difficulty (green < blue < black diamond)
Grade Percentage Explained
- 0% Grade: Completely flat, horizontal road
- 5% Grade: Climb 5 feet for every 100 feet horizontal (steep hill)
- 10% Grade: Climb 10 feet per 100 feet horizontal (very steep, near maximum for roads)
- 15% Grade: Climb 15 feet per 100 feet (extreme gradient, rarely in roads)
- 100% Grade: 45° angle (rise equals run)
Practical Examples
Driveway length 50 ft with 8 ft elevation change
Rise = 8 ft, Run = 50 ft
Slope = 8/50 = 0.16
Grade = 16%
Angle = arctan(0.16) ≈ 9.1°
Good for car access (not too steep)
Roof has 6" rise per 12" run
Slope = 6/12 = 0.5
Grade = 50%
Angle = arctan(0.5) ≈ 26.57°
This is a moderate roof pitch (good water drainage)
Point 1: (0, 0), Point 2: (3, 4)
Distance = √((3-0)² + (4-0)²) = √(9 + 16) = √25 = 5
Slope = 4/3 ≈ 1.33
This is a 3-4-5 right triangle!
Slope Categories
- Positive Slope: Line goes upward left to right (y increases as x increases)
- Negative Slope: Line goes downward left to right (y decreases as x increases)
- Zero Slope: Horizontal line (no vertical change)
- Undefined Slope: Vertical line (infinite rise, zero run)
Common Slope Ratios
- 1:12 (Wheelchair Ramp): Slope 0.083 (8.3%), Angle 4.76°, Safe accessible slope
- 1:8 (Handicap Ramp Steep): Slope 0.125 (12.5%), Angle 7.13°, Maximum ADA slope
- 1:4 (Moderate Incline): Slope 0.25 (25%), Angle 14.04°, Steep hiking trail
- 1:3 (Steep Slope): Slope 0.33 (33.3%), Angle 18.43°, Very steep terrain
- 1:1 (45° Angle): Slope 1.0 (100%), Angle 45°, Equal rise and run
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Frequently Asked Questions
1. What is slope?
Slope (m) measures steepness of a line. Formula: m = rise/run = (y₂-y₁)/(x₂-x₁). Example: 1/2 slope means for every 1 unit up, go 2 units over.
2. What's difference between slope and grade?
Slope is rise/run ratio (e.g., 1:2). Grade is same slope expressed as percentage (e.g., 50%). Grade % = (rise/run) × 100.
3. How do I find distance between two points?
Use distance formula: d = √((x₂-x₁)² + (y₂-y₁)²). This is Pythagorean theorem in coordinate plane.
4. What does negative slope mean?
Negative slope means line goes downward left to right. As x increases, y decreases. Example: slope -2 means go down 2 for every 1 right.
5. What's maximum safe slope for wheelchair ramp?
ADA standard: 1:12 ratio (8.3% grade, 4.76° angle). For shorter distances, 1:8 maximum (12.5%). Longer ramps need gentler slope.
6. How do I convert slope to angle?
Use arctangent: angle (degrees) = arctan(slope) × (180/π). Example: slope 1 = arctan(1) = 45°.
7. What's undefined slope?
Vertical line has undefined slope (infinite). Rise is finite but run is 0, so rise/run = undefined.
8. How is roof pitch measured?
Roof pitch is rise:run format (e.g., 6:12 means 6" rise per 12" run). Same as slope but uses inches.
9. What slope is safe for stairs?
Building code typically requires 30-35° angle (slope ~0.6-0.7). Too shallow feels odd, too steep is dangerous.
10. What's 45% grade in degrees?
45% grade = 0.45 slope = arctan(0.45) ≈ 24.2°. Not the same as 45° (which is 100% grade)!
11. How do slopes affect vehicles?
Most cars: < 10% (5.7°). Trucks: < 15% (8.5°). Steeper slopes cause traction loss, especially in snow/ice.
12. Can slope be greater than 1?
Yes! Slope > 1 means rise > run (steeper than 45°). Example: slope 2 = 45° angle with twice the rise of run.