How do you calculate slope from two points?

Given two points on a line, (x1, y1) and (x2, y2), the slope is calculated as (y2 − y1) divided by (x2 − x1), provided x2 is different from x1. If x2 equals x1 the line is vertical and the slope is undefined.

What is the relationship between slope, angle and percent grade?

If the slope is m, then the angle of the line with respect to the horizontal is the arctangent of m measured in degrees. The percent grade is 100 times the slope. For example, a slope of 0.1 corresponds to a 10 percent grade and an angle of about 5.71 degrees.

What does a zero or undefined slope mean?

A zero slope means the line is perfectly horizontal and there is no vertical change when moving along the line. An undefined slope occurs for a vertical line, where there is a change in y but no change in x. In that case the slope cannot be represented by a finite number.

Slope Calculator – From Two Points, Rise/Run, Angle, or Grade

What is the slope of a line?

In analytic geometry, the slope of a non-vertical line is the ratio between the vertical change and the horizontal change. It tells you how many units the line goes up or down when you move one unit to the right.

\[ m = \frac{\text{rise}}{\text{run}} \] If you know two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line, \[ m = \frac{y_2 - y_1}{x_2 - x_1}, \quad x_2 \neq x_1. \]

A positive slope means the line rises as you move to the right. A negative slope means it falls. A slope of zero corresponds to a horizontal line, while a vertical line has an undefined (infinite) slope because the run is zero.

Relationships between slope, angle and percent grade

If a line makes an angle \(\theta\) with the positive x-axis, measured in degrees, the slope and θ are related by

\[ m = \tan(\theta), \qquad \theta = \arctan(m). \]

In civil engineering and road design it is common to describe slopes as percent grade:

\[ \text{grade (\%)} = 100 \cdot m. \] For example, \(m = 0.08\) corresponds to an 8% grade.

Sign and magnitude of the slope

Sign: determines whether the line is increasing (positive), decreasing (negative), horizontal (zero), or vertical (undefined).
Magnitude: \(|m|\) measures how steep the line is. Large \(|m|\) means a steep line, small \(|m|\) means a gentle slope.

For many applications, it is useful to express the slope as a ratio such as “1:4” meaning 1 unit of rise for 4 units of run. This calculator reports both the decimal slope and a simplified rise:run ratio when possible.

FAQ – Slope and grade

Can a vertical line have a slope?

A vertical line has an undefined slope because the run (change in x) is zero. In the formula \(m = (y_2 - y_1) / (x_2 - x_1)\) the denominator would be zero, which cannot be represented by a finite real number. The calculator detects this case and labels the slope as vertical/undefined.

What is a gentle vs a steep slope?

There is no universal threshold, but in many practical contexts slopes with an absolute value less than about 0.05 (5%) are considered gentle, between 5–10% are moderate, and above 10% are steep. For roads and ramps there are often regulatory limits on maximum grades depending on usage and safety requirements.

How do I get the slope from the equation of a line?

If the equation is in slope–intercept form \(y = mx + b\), the coefficient of \(x\) is directly the slope. If the equation is in standard form \(Ax + By + C = 0\) with \(B \neq 0\), you can rearrange to \(y = -(A/B)x - C/B\), so the slope is \(-A/B\).

Can this calculator be used for incline ramps and roofs?

Yes. If you know the vertical rise and horizontal run of a ramp, stair, or roof, you can use the “Rise and run” mode to compute both the slope and the percent grade. This is useful for checking design guidelines or code requirements related to accessibility and safety.

1. Slope calculator