Linear Interpolation Calculator
Free Linear interpolation Calculator for coordinate geometry. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
y = y\u2081 + (x - x\u2081) \u00D7 (y\u2082 - y\u2081) / (x\u2082 - x\u2081)
Given two known points (x\u2081, y\u2081) and (x\u2082, y\u2082), the interpolated value y at position x is computed by proportionally blending between y\u2081 and y\u2082 based on where x falls relative to x\u2081 and x\u2082.
Worked Examples
Example 1: Temperature Estimation
Problem:At 8:00 AM the temperature was 60\u00B0F and at 12:00 PM it was 80\u00B0F. Estimate the temperature at 10:00 AM.
Solution:x1 = 8, y1 = 60, x2 = 12, y2 = 80, x = 10\ny = 60 + (10 - 8) * (80 - 60) / (12 - 8)\ny = 60 + 2 * 20 / 4\ny = 60 + 10 = 70\u00B0F\nFraction along: (10-8)/(12-8) = 0.50 (50% between points)\nSlope: 5\u00B0F per hour
Result:Temperature at 10:00 AM: 70\u00B0F | 50% along interval | Rate: 5\u00B0F/hour
Example 2: Engineering Table Lookup
Problem:Steel yield strength is 250 MPa at 20\u00B0C and 200 MPa at 400\u00B0C. Find strength at 150\u00B0C.
Solution:x1 = 20, y1 = 250, x2 = 400, y2 = 200, x = 150\ny = 250 + (150 - 20) * (200 - 250) / (400 - 20)\ny = 250 + 130 * (-50) / 380\ny = 250 - 17.105 = 232.895 MPa\nFraction along: (150-20)/(400-20) = 0.3421 (34.21%)
Result:Yield strength at 150\u00B0C: 232.9 MPa | 34.2% along interval
Frequently Asked Questions
What is the difference between interpolation and extrapolation?
Interpolation estimates values within the range of known data points (between x1 and x2), while extrapolation estimates values outside that range. Interpolation is generally reliable because the data points bound the estimate and the assumption of linearity is more reasonable over small intervals. Extrapolation, however, becomes increasingly unreliable as you move further from the known data because the linear assumption may not hold. A temperature measured between two weather stations can be interpolated with reasonable accuracy, but predicting temperature far beyond the station network is extrapolation and may produce unrealistic results. Linear Interpolation Calculator indicates whether your query falls within or outside the known range.
Where is linear interpolation used in practice?
Linear interpolation has widespread applications across many fields. In engineering, it is used to read between values in steam tables, material property tables, and calibration curves. In finance, it interpolates interest rates between benchmark maturities to construct yield curves. In computer graphics, it blends colors, positions, and textures between vertices (often called lerp). In scientific research, it estimates measurements between recorded observations. Weather services interpolate temperature, pressure, and humidity between weather stations. Manufacturing uses it for CNC machine tool paths between programmed points. Even everyday digital audio and image resizing rely heavily on linear interpolation.
How accurate is linear interpolation compared to other methods?
Linear interpolation is exact when the underlying relationship is truly linear, but introduces errors when the relationship is curved. The error is proportional to the square of the interval width and the magnitude of the second derivative (curvature) of the actual function. For smooth functions with small intervals between data points, linear interpolation is quite accurate. For curved functions, higher-order methods like cubic spline interpolation or polynomial interpolation provide better accuracy by capturing curvature. However, higher-order methods require more data points and can exhibit oscillation (Runge phenomenon) near the edges. For most practical engineering tables with reasonably spaced entries, linear interpolation achieves accuracy within 1-2% of the true value.
Can linear interpolation be used in multiple dimensions?
Yes, linear interpolation extends naturally to multiple dimensions. Bilinear interpolation works in two dimensions by performing three linear interpolations: first in the x-direction at two y-levels, then in the y-direction between those intermediate results. Trilinear interpolation extends this to three dimensions using seven linear interpolations. These multidimensional versions are essential in image processing (resizing and rotating images), 3D graphics (texture mapping), meteorology (spatial weather data), and scientific computing (interpolating simulation results on grids). Each additional dimension doubles the number of required data points: 2 for 1D, 4 for 2D, and 8 for 3D. The computational simplicity and reasonable accuracy make multidimensional linear interpolation the default choice in many applications.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy