Slope Calculator

Slope x
θ x
distance (d) x
ΔX x
ΔY x

(x1,y1) =

(x2,y2) =





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Slope Calculator

Slope is a concept that finds multiple usages in real-time. On this page, let us learn in deep the various concepts that are associated with slope and its calculation, its applications in real-time and some fascinating facts about slope.

What is the slope?

Slope is the steepness of a line with respect to the x-axis. In real-time, if you consider x-axis as the ground, slope shows how risen is the way or path or an object with respect to the ground.

How to calculate slope?

Consider two points (x1,y1) and (x2,y2) in a two dimensional space. Then the line joining these two points has a slope given by the following formula w.r.t to the x-axis.

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

Significance of Slope

Though we all know slope as a mathematical concept, it is increasingly used in many applications of physics, geographical studies, and sports. Before we get into the calculation of slope, it is hence important to understand the diversified usage of slope in various aspects of our life.

Roads and Travelling

It is a common sight to all of us that land is never in a straight form. We come across ups and downs and steep areas many times during our journeys or in day to day life too. You must also have noticed road signs showing “Steep Hill Ahead” or “20% with a steep triangle image”. These are all nothing but the signs that alert you that the road ahead is steep and that you should drive carefully.

Laying of roads and allotting permissions for vehicles in such places is based on such studies only.

Landscaping and Farming

Hill stations and steep landforms offer a great sight to the viewers. However, there exist many pros and cons because of the slope. Water runoff is a determining factor to estimate which crops suit well that landforms and the cost of farming and measures to be taken for better productivity.

Geographers understand and classify soils, differentiate landforms and study the model surface runoff

Geographic Information Systems (GIS)

The slope of a landform plays a key role in Geographic Information Systems (GIS) to analyze elevation points, to stimulate and design elevation models and to study contour lines.

Economics

Economists use the concept of slope to show how things are changing. For instance, the famous Philips curve that shows the variation of unemployment and inflation is a downward sloping curve. Similarly, the demand and price curve also is slopy and shows how the two terms are related to one another.

Mathematics

Slope is an indispensable concept in linear algebra and to describe line gradients.

The slope-intercept form of the line is well known in linear algebra. The usual notation of the slope-intercept form of the line is : y=mx+c, where m is the slope of the line, which we calculate by the formula {(y2-y1)/(x2-x1)} or dy/dx, as it is popularly called in short.

Mathematically, you can interpret slope in this way too: The change in y or the dependent variable with the unit change in x or the independent variable.

Do you know?

  • Slope represents not just steepness; it also effectively tells the direction. Yes.
  • When moving from left to right, if the line has a positive slope, it means the line is moving upwards. In normal terms, you can consider this as an “uphill” scenario.
  • Similarly, when moving from left to right, if the line has a negative slope, it means the line is going downwards. This can be considered as “downhill” scenario in real life.

What is meant by Infinite Slope?

From the slope formula, we obtain an infinite slope when dx=0. It means there is no change in x-variable. Then in such a situation, we get nothing but a straight line passing through a fixed point on x-axis parallel to the y-axis. Thus a line with infinite slope is nothing but y-axis or any line parallel to the y-axis.

Interesting facts about slope

  • A steep slope is different from a gentle slope, especially when you are considering to cultivate crops. A land with steep slope experiences faster soil erosion compared to the latter one.
  • Lands that have a slope on the north are wetter compared to those on the south. This is because of the sun’s rays that evaporate moisture of south-facing slope lands quicker.
  • Skiers and snowboarders always consider slope and the related impacts to assess potential dangers associated while skiing.
  • The wheelchair ramps, loading ramps and steps in a building are designed keeping in mind the slope required for a given weight on a particular surface.
  • In art and paintings, the slope of lines and sketches plays a huge impact in creating aesthetic sense to our eye.
  • Most roads are given a slight slope towards the edges to ensure that rainwater is easily drained.
  • While designing the slope of a roof, architects take into account the prescribed slope limits as set by the local state rules.
  • The letter 'm’ in the slope-intercept where 'm’ is the common notation for slope came into existence from a French word ‘monte’ which means ‘climb’ or ‘slope’ in English.
  • The word Slope is derived from the Latin root span which means ‘slip’.
  • While calculating slope, it doesn’t matter which coordinates you take as the starting point; it means even if you interchange x1 with x2 and y1 with y2, the slope remains the same but never gets reversed!

Fun Fact

The usual notation of slope is y-mx+c. However, this is not standard too. There exist many other forms for a slope-intercept form of a line.

For example:

  • y = kx + m, as per Swedish textbooks, where "slope" is called "riktningskoefficient", meaning "direction coefficient”.
  • Y=kx+d, as per Austria textbooks.
  • y = ax + b or px + q or mx + n, as per Netherlands textbooks.
  • y = ax + b or y = mx + n, as per Uruguay textbooks.

Life is more delightful on the downward slope. – Seneca the Younger.

How CalculatorHut’s slope calculator helps you?

CalculatorHut, the expert in simplifying calculations and delivering online calculators for free, offers the free online slope calculator. This is simple, user-friendly and accurate to many decimals. You can calculate the slope of any line related to any coordinates in a jiffy!

For many students, who fear math calculations, the wide range of free scientific and math calculators present on CalculatorHut are very handy. They can use it for quick calculations or for comparing the results of their calculations. The site is well designed with a relevant explanation for each topic for the calculation to let the user have conceptual clarity too. Learner friendly, isn’t it?

If you are an enthusiast of calculations or own a website or a blog and wish to embed any calculator from the CalculatorHut as a widget, you can drop us a word at calculatorhut@gmail.com. We will design a free widget as per your customizations.

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Aggression is the first step on the slippery slope to selfishness and chaos. – Anne Campbell

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