# Slope Calculator

Created by Mateusz Mucha and Julia Żuławińska
Reviewed by Bogna Szyk and Jack Bowater
Last updated: Dec 21, 2022

The slope calculator determines the slope or gradient between two points in the Cartesian coordinate system. The slope is basically the amount of slant a line has and can have a positive, negative, zero, or undefined value. Before using the calculator, it is probably worth learning how to find the slope using the slope formula. To find the equation of a line for any given two points that this line passes through, use our slope intercept form calculator.

## How to find slope

1. Identify the coordinates $(x_1, y_1)$ and $(x_2, y_2)$. We will use the formula to calculate the slope of the line passing through the points $(3, 8)$ and $(-2, 10)$.
2. Input the values into the formula. This gives us $(10 - 8)/(-2 - 3)$.
3. Subtract the values in parentheses to get $2/(-5)$.
4. Simplify the fraction to get the slope of $-2/5$.
5. Check your result using the slope calculator.

To find the slope of a line, we need two coordinates on the line. Any two coordinates will suffice. We are basically measuring the amount of change of the y-coordinate, often known as the rise, divided by the change of the x-coordinate, known the run. The calculations in finding the slope are simple and involve nothing more than basic subtraction and division.

🙋 To find the gradient of non-linear functions, you can use the average rate of change calculator.

## The slope formula

$\mathrm{slope} = \frac{y_2 - y_1}{x_2 - x_1}$

Notice that the slope of a line is easily calculated by hand using small, whole number coordinates. The formula becomes increasingly useful as the coordinates take on larger values or decimal values.

It is worth mentioning that any horizontal line has a gradient of zero because a horizontal line has the same y-coordinates. This will result in a zero in the numerator of the slope formula. On the other hand, a vertical line will have an undefined slope since the x-coordinates will always be the same. This will result the division by zero error when using the formula.

Just as slope can be calculated using the endpoints of a segment, the midpoint can also be calculated. The midpoint is an important concept in geometry, particularly when inscribing a polygon inside another polygon with its vertices touching the midpoint of the sides of the larger polygon. This can be obtained using the midpoint calculator or by simply taking the average of each x-coordinates and the average of the y-coordinates to form a new coordinate.

The slopes of lines are important in determining whether or not a triangle is a right triangle. If any two sides of a triangle have slopes that multiply to equal -1, then the triangle is a right triangle. The computations for this can be done by hand or by using the right triangle calculator. You can also use the distance calculator to compute which side of a triangle is the longest, which helps determine which sides must form a right angle if the triangle is right.

The sign in front of the gradient provided by the slope calculator indicates whether the line is increasing, decreasing, constant or undefined. If the graph of the line moves from lower left to upper right it is increasing and is therefore positive. If it decreases when moving from the upper left to lower right, then the gradient is negative.

## FAQ

### How to find slope from an equation?

The method for finding the slope from an equation will vary depending on the form of the equation in front of you. If the form of the equation is y = mx + c, then the slope (or gradient) is just m. If the equation is not in this form, try to rearrange the equation. To find the gradient of other polynomials, you will need to differentiate the function with respect to x.

### How do you calculate the slope of a hill?

1. Use a map to determine the distance between the top and bottom of the hill as the crow flies.
2. Using the same map, or GPS, find the altitude between the top and bottom of the hill. Make sure that the points you measure from are the same as step 1.
3. Convert both measurements into the same units.
4. Divide the difference in altitude by the distance between the two points.
5. This number is the gradient of the hill if it increases linearly. If it does not, repeat the steps but at where there is a noticeable change in slope.

### How do you calculate the length of a slope?

1. Measure the difference between the top and bottom of the slope in relation to both the x and y axis.
2. If you can only measure the change in x, multiply this value by the gradient to find the change in the y axis.
3. Make sure the units for both values are the same.
4. Use the Pythagorean theorem to find the length of the slope. Square both the change in x and the change in y.
5. Add the two values together.
6. Find the square root of the summation.
7. This new value is the length of the slope.

### What is a 1 in 20 slope?

A 1/20 slope is one that rises by 1 unit for every 20 units traversed horizontally. So, for example, a ramp that was 200 ft long and 10 ft tall would have a 1/20 slope. A 1/20 slope is equivalent to a gradient of 1/20 (strangely enough) and forms an angle of 2.86° between itself and the x-axis.

### How do you find the slope of a curve?

As the slope of a curve changes at each point, you can find the slope of a curve by differentiating the equation with respect to x and, in the resulting equation, substituting x for the point at which you’d like to find the gradient.

### Is rate of change the same as slope?

The rate of change of a graph is also its slope, which are also the same as gradient. Rate of change can be found by dividing the change in the y (vertical) direction by the change in the x (horizontal) direction, if both numbers are in the same units, of course. Rate of change is particularly useful if you want to predict the future of previous value of something, as, by changing the x variable, the corresponding y value will be present (and vice versa).

### Where do you use slope in everyday life?

Slopes (or gradients) have a number of uses in everyday life. There are some obvious physical examples - every hill has a slope, and the steeper the hill, the greater its gradient. This can be useful if you are looking at a map and want to find the best hill to cycle down. You also probably sleep under a slope, a roof that is. The slope of a roof will change depending on the style and where you live. But, more importantly, if you ever want to know how something changes with time, you will end up plotting a graph with a slope.

### What is a 10% slope?

A 10% slope is one that rises by 1 unit for every 10 units travelled horizontally (10 %). For example, a roof with a 10% slope that is 20 m across will be 2 m high. This is the same as a gradient of 1/10, and an angle of 5.71° is formed between the line and the x-axis.

### How do you find the area under a slope?

To find the area under a slope you need to integrate the equation and subtract the lower bound of the area from the upper bound. For linear equations:

1. Put the equation into the form y = mx + c.
2. Write a new line where you add 1 to the order of the x (e.g., x becomes x2, x2.5 becomes x3.5).
3. Divide m by the new number of the order and put it in front of the new x.
4. Multiply c by x and add this to the new line.
5. Solve this new line twice, one where x is the upper bound of the area you wish to find and one where x is the lower bound.
6. Subtract the lower bound from the upper bound.
7. Congratulate yourself on your achievement.

### What degree is a 5 to 1 slope?

A 5 to 1 slope is one that, for every increase of 5 units horizontally, rises by 1 unit. The number of degrees between a 5 to 1 slope and the x-axis is 11.3°. This can be found by first calculating the slope, by dividing the change in the y direction by the change in the x direction, and then finding the inverse tangent of the slope.

Mateusz Mucha and Julia Żuławińska
First point coordinates
x₁
y₁
Second point coordinates
x₂
y₂
Result
Slope (m)
Related numbers
Y - intercept
Angle (θ)
deg
%
Distance (d)
Distance between x's (Δx)
Distance between y's (Δy)
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