# Definition:Slope

## Definition

### Straight Line

Let $\LL$ be a straight line embedded in a Cartesian plane.

The slope of $\LL$ is defined as the tangent of the angle that $\LL$ makes with the $x$-axis.

### Curve

Let $P$ be a point on a curve $\CC$ embedded in a Cartesian plane.

The slope of $\CC$ at $P$ is defined as the slope of the tangent to $\CC$ at $P$.

## Also defined as

### Also defined as: $1$

Slope can also be seen defined as the angle the line makes with the horizontal.

### Also defined as: $2$

In the context of geography, the slope of a road is often seen defined as the vertical distance travelled with respect to the actual distance travelled along the road.

## Examples

### Arbitrary Example

A slope or gradient of $1$ in $4$ indicates a vertical distance of $1$ unit for every $4$ units travelled along the road.

This is also indicated as a ratio $1 : 4$ or a fraction $1 / 4$, or can also be expressed as the angle of slope, that is, $\arcsin \dfrac 1 4$ or a percentage, that is, a $25 \%$ gradient.

## Also known as

The slope of a straight line or curve is also sometimes referred to as its gradient.

However, that term has a more generic and abstract meaning than does the concept of slope as given here.

The word grade can sometimes be seen, but this is discouraged as it has a number of meanings.

Some sources suggest that the slope of a straight line is the same as its direction, but this is true only in the plane.

## Also see

• Results about slope can be found here.