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By definition, the slope or gradient of a line describes its steepness, incline, or grade. <>< p=' />m — slope < />θ — angle of incline |
If the 2 Points are Known
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If 1 Point and the Slope are Known
Slope, sometimes referred to as gradient in mathematics, is a number that measures the steepness and direction of a line, or a section of a line connecting two points, and is usually denoted by m. Generally, a line's steepness is measured by the absolute value of its slope, m. The larger the value is, the steeper the line. Given m, it is possible to determine the direction of the line that m describes based on its sign and value:
- A line is increasing, and goes upwards from left to right when m > 0
- A line is decreasing, and goes downwards from left to right when m < 0
- A line has a constant slope, and is horizontal when m = 0
- A vertical line has an undefined slope, since it would result in a fraction with 0 as the denominator. Refer to the equation provided below.
Slope is essentially change in height over change in horizontal distance, and is often referred to as 'rise over run.' It has applications in gradients in geography as well as civil engineering, such as the building of roads. In the case of a road the 'rise' is the change in altitude, while the 'run' is the difference in distance between two fixed points, as long as the distance for the measurement is not large enough that the earth's curvature should be considered as a factor. The slope is represented mathematically as:
| m = |
|
In the equation above, y2 - y1 = Δy, or vertical change, while x2 - x1 = Δx, or horizontal change, as shown in the graph provided. It can also be seen that Δx and Δy are line segments that form a right triangle with hypotenuse d, with d being the distance between the points (x1, y1) and (x2, y2). Since Δx and Δy form a right triangle, it is possible to calculate d using the Pythagorean theorem. Refer to the Triangle Calculator for more detail on the Pythagorean theorem as well as how to calculate the angle of incline θ provided in the calculator above. Briefly:
d = √(x2 - x1)2 + (y2 - y1)2
The above equation is the Pythagorean theorem at its root, where the hypotenuse d has already been solved for, and the other two sides of the triangle are determined by subtracting the two x and y values given by two points. Given two points, it is possible to find θ using the following equation:
m = tan(θ)
Given the points (3,4) and (6,8) find the slope of the line, the distance between the two points, and the angle of incline:
| m = |
| = |
|
d = √(6 - 3)2 + (8 - 4)2 = 5
| = tan(θ) |
| θ = tan-1( |
| ) = 53.13° |
While this is beyond the scope of this calculator, aside from its basic linear use, the concept of a slope is important in differential calculus. For non-linear functions, the rate of change of a curve varies, and the derivative of a function at a given point is the rate of change of the function, represented by the slope of the line tangent to the curve at that point.
Rectangular Prism Shape
l = length
w = width
h = height
d = diagonal
Stot = total surface area
Slat = lateral surface area
Stop = top surface area
Sbot = bottom surface area
V = volume
Calculator Use
Enter any 3 variables for a rectangular prism into this online calculator to calculate the other 3 unknown variables. A cube is a special case where l = w = h for a rectangular prism. Elmo driver download for windows.
Units: Note that units are shown for convenience but do not affect the calculations. The units are in place to give an indication of the order of the results such as ft, ft2 or ft3. For example, if you are starting with mm and you know h, l and w in mm, your calculations will result with d in mm, S in mm2 and V in mm3.
Formulas for a rectangular prism:
- Volume of Rectangular Prism:
- V = lwh
- Surface Area of Rectangular Prism:
- S = 2(lw + lh + wh)
- Space Diagonal of Rectangular Prism: (similar to the distance between 2 points)
- d = √(l2 + w2 + h2)
A cube is a special case where l = w = h. So you can find the volume of a cube or surface area of a cube by setting these values equal to each other.
Calculations for a rectangular prism:
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1. Given the length, width and height find the volume, surface area and diagonal of a rectangular prism
- h, l and w are known; find V, S and d
- V = lwh
- S = 2(lw + lh + wh)
- d = √(l2 + w2 + h2)
2. Given the surface area, length and width find the height, volume and diagonal of a rectangular prism
- S, l and w are known; find h, V and d
- h = (S - 2lw) / (2l + 2w)
- V = lwh
- d = √(l2 + w2 + h2)
3. Given the volume, length and width find the height, surface area, and diagonal of a rectangular prism
- V, l and w are known; find h, S and d
- h = V / lw
- S = 2(lw + lh + wh)
- d = √(l2 + w2 + h2)
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4. Given the diagonal, length and width find the height, volume and surface area of a rectangular prism Giesecke & devrient network & wireless cards driver download for windows 10.
- d, l and w are known; find h, V and S
- h = √(d2 - l2 - w2)
- V = lwh
- S = 2(lw + lh + wh)
For more information on cuboids see: Weisstein, Eric W. 'Cuboid.' From MathWorld--A Wolfram Web Resource, Cuboid.