This calculator performs linear interpolation between two known points and determines the intermediate value of Y for a given X. This type of calculation is used when the pairs of values (X1, Y1) and (X2, Y2) are known, and an estimated value is needed between them under a straight-line relationship.
The method is used with engineering tables, graphs, reference data for materials, temperature relationships, loads, flow rates, coefficients, and other quantities where a straight-line approximation is acceptable over the selected interval. The calculator also displays the result on a graph so the position of the target point can be seen relative to the original data.
Basis of the method is that the quantity is assumed to change linearly between two known points. This means that when X changes by equal steps, the value of Y changes uniformly over the selected interval.
Y = Y1 + (X - X1) / (X2 - X1) × (Y2 - Y1)
Meaning of the formula is as follows. First, the method determines what fraction of the distance along the X axis the point X occupies between X1 and X2. Then the same fraction is applied to the difference Y2 - Y1. After that, the resulting increment is added to the starting value Y1.
Step 1 - the input data provide two coordinates on the X axis and two corresponding values on the Y axis. The units may be any units, but they must be consistent on each axis. For example, if X is given in °C, then X1 and X2 must also be in °C. The same principle applies to Y.
Step 2 - the calculator finds the difference X - X1 and the total interval X2 - X1. The ratio of these values shows the relative position of the target point on the X axis.
Step 3 - the change along the Y axis is calculated as Y2 - Y1. This difference is then multiplied by the previously found fraction of the interval along X.
Step 4 - the resulting increment is added to Y1. The result is the calculated value of Y in the same unit as Y1 and Y2.
Interpolation is valid when the target value X lies between X1 and X2. In this case, the result is an intermediate value on the segment between the two known points.
Extrapolation occurs if X is smaller than X1 or greater than X2. Mathematically, the formula remains the same, but the result falls outside the original interval. In practice, this estimate is less reliable because the real relationship outside the known range may no longer be linear.
Boundary case with X1 = X2 is not allowed because the denominator X2 - X1 becomes 0. In that case, the formula cannot be used.
Accuracy of the method depends not on the number of decimal places, but on how closely the real relationship follows a straight line over the selected interval. The shorter the interval between X1 and X2, the more often linear interpolation gives a stable result.
Common approach is to apply the method to tabulated data where neighboring points are already close enough to each other. If the step between the original values is large and the relationship is clearly non-linear, the result may only be an approximation.
Logic check is simple. If X is located exactly halfway between X1 and X2, then under a linear relationship the value of Y should also be exactly halfway between Y1 and Y2. This is a quick way to verify the calculation visually.
Symbols and formula notation in engineering calculations are commonly written in line with ISO 80000-2:2019 "Quantities and units - Part 2: Mathematics", which sets general rules for mathematical symbols and notation of expressions.
Engineering use of the method appears when working with tabulated and graphical data in calculations under the Eurocodes. In particular, EN 1990 "Eurocode - Basis of structural and geotechnical design" provides the general calculation framework for engineering verification, and linear interpolation in such tasks is used as an auxiliary numerical method for estimating intermediate values between known points.
It works best where the relationship between two neighboring points is close to a straight line. For engineering tables and reference graphs, this is usually acceptable over short intervals where the parameter changes smoothly.
Mathematically, the formula will still calculate a result, but this is no longer interpolation. It becomes linear extrapolation. For practical tasks, such an answer should be treated with caution because the real curve outside the known range may behave differently.
This is because the formula contains division by X2 - X1. If that difference equals zero, it is impossible to determine the relative position of the point on the X axis, and the calculation loses its meaning.
Yes. Within each axis, the units must be consistent. The values X, X1, and X2 must be entered in one common unit, while Y, Y1, and Y2 must be entered in one consistent unit for that quantity.
Linear interpolation does not reconstruct the original law of variation of the quantity. It approximates that relationship by a straight line between two known points. Because of that, it is not a universal replacement for an analytical formula, but a practical way to obtain an intermediate value quickly from a table or graph.