CalCon has developed a Gradient calculator, a tool for calculating a gradient. Do you know what gradient is? Sounds interesting, doesn’t it? With this calculator you will learn how to compute the gradient of a line that passes through two locations, what are gradient properties, how you can calculate gradient in the middle of forest and so much more.

Furthermore, this article says briefly about what kind of gradients we have, how how to calculate gradient with special formula, and how to calculate it step by step. Later on, we will do a few examples for practice.

While you are here, check out our other math calculators, such as Spherical Coordinates Calculator and Tangent Calculator.

Gradient (slope) in math – Definition

The slope (m) of a curve is another term for the gradient. For example, the tangent of an angle is equal to the slope or gradient of a plane inclined at that angle. Also, the sharper the line is at a place where the gradient of a graph is higher. A negative gradient indicates a descending slope. Moreover, we can determine the gradient geometrically for any two points on a line (x1,y1), (x2,y2). Furthermore, the gradient is a variable that aids in comprehending variability in one quantity concerning another. Moreover, the gradient of a function f(x) is determined using its first derivative:

\frac{d}{dx} {f}{(x)}

To summarize the preceding sentences, we have:


We use “vertical shift” to “horizontal shift” ratio among (any) two unique points on a line to compute slope. We can write sometimes the ratio as a divisor (“rise over run”). It also gives the same value for every two different points on a single line. The “rise” of a falling line is negative. A road surveyor may draw the line, or it may appear in a schematic that depicts a road.

Advanced mathematics specifies the gradient of a line at a spot as the gradient of the tangent line at that point. When the line is represented by a sequence of dots in a schematic or a list of point coordinates, it can be determined between any two specified points. Whenever we represent the line as a continuous function, such as an algebratic formula, the differential calculus gives rules that yield an equation for the portion of the curve.

Mathematicians commonly use a simple fraction to express a gradient. Also they measure it as a proportion or a decimal value.

Gradient properties

This is a list of the features of a gradient to assist you in comprehending the line’s orientation:

  • The value of a gradient might be potentially positive or negative.
  • The x-axis gradient is 0 because the slope of a horizontal line is zero.
  • The y-axis gradient is indeterminate because the slope of a vertical line is undefined.
  • The slope of a curve at every point on the curve is the same as a slope of its tangent at a certain point.
  • Two parallel lines have the same gradient. m1=m2
  • And finally the product of two perpendicular lines’ gradients is -1.

Types of Gradient

There are six types of gradient:

  • Ruling gradient
  • Limiting gradient
  • Exceptional gradient
  • Minimum gradient
  • Average gradient
  • Floating gradient

Gradient Formula

To compute gradient or slope, the ratio of the rise (vertical change) over to run (horizontal change) must be computed between two points on the line. Thereofore you can do it with this formula:


Gradient (slope) calculation – step by step

You need two points from a line to determine its slope: (x1, y1) and (x2, y2). The formula for calculating the slope between two points is:


But, when you don’t have the equation for a straight line, there are three parts to calculate its slope:

  • Firstly, identify two points: a starting place and a destination.
  • Determine whether there is a positive or negative gradient.
  • We can express the gradient as a percentage. The abrupt change appears at the top, whereas the horizontal change appears at the bottom. It’s possible to write it as:

The gradient of a line

We choose two places on the line to determine the slope of a straight line. Firstly, we make a calculation based on these two points: The height difference (y coordinates) divided by The width difference (x coordinates). Furthermore, if the solution is a positive number, the line goes uphill. If the response is negative, the line is going downhill.

Also, a line’s gradient refers to how steep a straight line is. The gradient is symbolized by the character m in the given formula of a horizontal path,

y=m\cdot x + c

Gradients might be positive or negative, but we always observe them clockwise.

Slope (inclination) chart

On a normal graph, the x– and y-axes are perpendicular and produce four right angles. The inclination of a graph with only x and y lines is always 90 degrees. This is because we define the inclination as the positive section of the x-axis (the two vertical quadrants of a graph) until it reaches a line. Because the y-axis is the only other line on the graph, inclination spans the entire upper right quadrant, making it 90 degrees. Any horizontal line has a rise of 0 as well as any vertical line has a rise of 90. It’s worth nothing that horizontal lines reflect the x-axis, while vertical lines reflect the y-axis.

Gradient vector

We can look at the “direction and rate of fastest increase” as the gradient vector. Whenever the path of the slope is the path where the function rises fastest from p, and the magnitude of the slope is the rate of growth in that direction, the highest absolute directional derivative if the slope of a function is non-zero at a point p.

Furthermore, if and only if a point is stationary, the gradient is the zero vector (where the derivative vanishes). As a result, the gradient is important in optimization theory, in which using gradient ascent results in function maximization.

Slope Conversion Table

Horizontal to Vertical slope ratio 1/41/23/411-1/41-1/21-3/422-1/23456789101112131415161718
Percent slope400200133100806757504433252016.714.312.511.
Degrees slope76.063.453.145.038.733.729.726.624.018.414.
Inches per Foot482416129.686.965.3432.

Gradient Calculator – Example

To compute the gradient, we must first locate two points in Cartesian coordinates (x1, y1) and (x2, y2). Firstly, let’s imagine we want to find the slope of a line that passes through a set of points (5, 2) and (3, 7).

  • Enter the first point’s parameters as x1 and y1 in the calculator.
  • Now do the same with the second point’s parameters. Just switch to x2 and y2.
  • And last but not least, the calculator will calculate the slope equation and count it for you automatically.

Gradient calculator – Practical examples

United Kingdom use ratios to describe steepness on road signs. In this case, the road sign displays a 1:3 ratio. The first value refers to the change in vertical distance, while the second value relates to the distance traveled horizontally. When compared to a fully flat road, a 1:3 ratios indicate that for every three units horizontally traveled, the road is one unit higher up. In conclusion we express this as a slope of 1/3 in a math class.


What does gradient mean in maths?

Mathematicians use the gradient as a differential operator in mathematics for a three-dimensional vector-valued function to produce a vector with three components which are also the fractional derivatives of the function concerning its three parameters.

How to calculate gradient on a topographic map?

When calculating a slope on a topographic map, it’s important to understand that the terms “gradient” and “slope” are equivalent. This is because the pattern of the land reveals itself by the gradient shift that occurs inside a certain location on the map. As a result, geologists and environmentalists can establish whether the gradient of the designated area affects the surrounding areas. Furthermore, erosion is a fantastic example of why it’s critical to understand the slope of specific locations.

How to calculate stream gradient?

The gradient of the stream’s channel is referred to as stream gradient. It is the stream’s vertical drop over a horizontal distance. We can use the following equation to compute it: Gradient=\frac{change in elevation}{distance}
We commonly represent it in feet per mile or meters per kilometer. Stream gradients are often stronger in the headwaters (where the stream starts) and lower at the mouth (where it empties into some other body of water) (such as the ocean).

How is gradient calculated in geography?

Use vertical data to compute height difference between points, then use the scale on your map to measure the horizontal distance from point A to B. And finally, plug all the information into the formula to compute it. Formula is: Gradient=\frac{difference in height}{horizontal distance}

What is the formula of velocity gradient?

The term “velocity gradient” refers to calculating velocity per unit distance. Finally, we can describe the equation for velocity gradient as: Velocity \; gradient=\frac{velocity}{distance}

How do you find the gradient of a road?

The pace at which the road level rises or falls concerning the horizontal distance along its length. In some countries the following things influence the pavement’s gradient:
Traffic characteristics;
Drainage, safety, appearance, and access to neighboring land are all physical aspects of the site;
And finally, line-intersection, bridge, approach road, and railway.

How do you calculate climb gradient?

You can compute the rate of climb manually if a climb gradient table is not available. Whenever you want to compute it, just multiply your ascent gradient in feet per nautical mile by your ground speed in nautical miles per hour, divided by 60 minutes per hour. The required rate of ascent in feet per minute will be the outcome.