Gradient (Slope) Calculator

Calculate the gradient/slope between two points. Find rise, run, angle of inclination, distance, and classify the slope type.

Enter Two Points

Point 1 (x1, y1)
Point 2 (x2, y2)

Result

Gradient (Slope)
2
Slope (m) 2
Rise (dy) 8
Run (dx) 4
Angle of Inclination 63.4349
Distance Between Points 8.944272
Midpoint (3, 6)
y-intercept (b) 0
Line Equation y = 2x + 0
Slope Type Positive
Percentage Grade 200%

Step-by-Step Solution

m = (y2 - y1) / (x2 - x1) = (10 - 2) / (5 - 1) = 8/4 = 2

Understanding Gradient (Slope)

The gradient, or slope, of a line measures its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. The slope is a fundamental concept in algebra, calculus, physics, and engineering.

Slope Formulas and Related Concepts

Slope Formula

The ratio of rise over run between two points.

m = (y2 - y1) / (x2 - x1)

Slope-Intercept Form

The equation of a line with slope m and y-intercept b.

y = mx + b

Point-Slope Form

Line equation through a specific point.

y - y1 = m(x - x1)

Angle of Inclination

The angle the line makes with the positive x-axis.

theta = arctan(m)

Distance Formula

The distance between two points in a plane.

d = sqrt((x2-x1)^2 + (y2-y1)^2)

Midpoint Formula

The point exactly halfway between two points.

M = ((x1+x2)/2, (y1+y2)/2)

Types of Slope

  • Positive slope (m > 0): Line rises from left to right. The larger the value, the steeper the climb.
  • Negative slope (m < 0): Line falls from left to right. More negative means steeper descent.
  • Zero slope (m = 0): Horizontal line. No vertical change between any two points.
  • Undefined slope: Vertical line. The run (dx) is zero, making the fraction undefined.

Practical Applications

Slope calculations are used in many real-world contexts:

  • Civil engineering: Road grades, ramp slopes, and drainage gradients must meet specific slope requirements.
  • Physics: Velocity-time graphs use slope to represent acceleration; position-time graphs show velocity.
  • Economics: Marginal cost and revenue are slopes of their respective curves.
  • Geography: Terrain slope affects water runoff, erosion, and land use planning.
  • Construction: Roof pitch, stair rise/run ratios, and wheelchair ramp grades all involve slope.

Percentage Grade

Percentage grade expresses slope as a percentage: grade = (rise/run) x 100%. A 100% grade means a 45-degree angle. Highway grades rarely exceed 6-8%. The steepest public roads in the world approach 35% grade.