Last updated: July 31, 2026
Linear Interpolation Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
y = y0 + (x - x0) × (y1 - y0) / (x1 - x0), t = ((x - x0) / (x1 - x0)) × 100%
Where:
- x₀=First known x-value
- y₀=Output at x₀
- x₁=Second known x-value
- y₁=Output at x₁
- x=Target x-value
- y=Interpolated output
- t=Interpolation position as a percentage
Worked Examples
Estimate between two rising points
Use two known points on a line to estimate an interior value.
- 1Compute the slope: (50 - 10) / (20 - 0) = 2
- 2Measure the horizontal change: 5 - 0 = 5
- 3Add the change to y₀: 10 + 5 × 2 = 20
- 4Compute t = (5 / 20) × 100 = 25%
Decimal interpolation
Linear interpolation also works cleanly with decimal coordinates.
- 1Slope = (20 - 4) / (6 - 2) = 4
- 2Horizontal shift = 3.5 - 2 = 1.5
- 3Interpolated y = 4 + 1.5 × 4 = 10
- 4t = (1.5 / 4) × 100 = 37.5%
Extrapolation beyond the second point
If x falls outside the interval, the same line formula produces an extrapolated result.
- 1Slope = (30 - 0) / (10 - 0) = 3
- 2Horizontal shift = 12 - 0 = 12
- 3Interpolated/extrapolated y = 0 + 12 × 3 = 36
- 4t = (12 / 10) × 100 = 120%
Invalid when x₀ equals x₁
The formula is undefined if both known points share the same x-coordinate.
- 1Check the denominator x₁ - x₀
- 2Here the denominator is 5 - 5 = 0
- 3Division by zero makes the interpolation undefined
- 4Choose two points with different x-values
Introduction
The Linear Interpolation Calculator estimates a y-value on the straight line passing through two known points. It is useful when you know measurements, prices, temperatures, or engineering values at two x-locations and need a quick estimate in between. The calculator also shows the interpolation factor as a percentage, so you can see exactly where the target x sits relative to the original interval.
What Linear Interpolation Does
Linear interpolation assumes the change between two known points is straight and uniform. That makes it one of the fastest and most transparent estimation tools in algebra, spreadsheets, chart reading, and engineering approximations.
Uses exactly two known points
Builds the straight-line equation between them
Estimates y for any target x-value
Reports position within the interval as a percentage
Can also extend outside the interval for extrapolation
Interpolation is usually most reliable when x lies between x₀ and x₁.
Formula Breakdown
The interpolation formula starts with y₀, then adds the horizontal change scaled by the slope between the two points. The factor t measures the relative horizontal position and turns that proportion into a percentage.
Slope = (y₁ - y₀) / (x₁ - x₀)
Horizontal change = x - x₀
Scaled rise = (x - x₀) × slope
Interpolated output = y₀ + scaled rise
t = 0% at x₀ and t = 100% at x₁
How to Use This Calculator
Enter the two known points and then enter the target x where you want the estimate. The calculator instantly returns the interpolated y and the factor t so you can verify whether the target is inside the original range or outside it.
Enter x₀ and y₀ for the first point
Enter x₁ and y₁ for the second point
Enter the target x-value
Read the computed y-value
Check t to understand the relative position
Understanding the Percentage Factor
The t output is more than a helper value. It tells you where the target x lies on the line segment when measured from x₀ toward x₁, which is especially helpful in reporting, calibration, and quality-control workflows.
0% means x equals x₀
50% means x is halfway between x₀ and x₁
100% means x equals x₁
Negative values mean x lies before x₀
Values above 100% mean extrapolation beyond x₁
Validation Rules
Interpolation only works when every input is a finite number and the denominator is not zero. If x₀ and x₁ are equal, the line would be vertical, so the formula for y as a function of x is undefined.
All five inputs must be numeric and finite
x₀ and x₁ must be different
Decimal and negative values are allowed
The method supports interpolation and extrapolation
Undefined cases return a clear error message
A vertical line cannot be expressed with the given y(x) formula.
Common Mistakes to Avoid
Most wrong answers come from swapping coordinates, using inconsistent units, or treating nonlinear data as if it were linear. A quick check of the slope and the t value usually catches these issues early.
Swapping x-values and y-values
Using x-values with different units
Assuming curved data is linear
Ignoring that t above 100% is extrapolation
Forgetting to verify x₁ - x₀ before calculating
Practical Applications
Linear interpolation appears anywhere people estimate a value between two measured points. It is common in finance, sensor calibration, property tables, statistics, digital graphics, and scientific tables.
Estimating values from charts or lookup tables
Interpolating temperatures or pressures
Approximating prices between breakpoints
Calibrating instruments from test data
Checking spreadsheet or code outputs quickly
FAQs
What is linear interpolation?
It is a method for estimating a value between two known points by assuming the relationship between them is a straight line.
What does the t output mean?
t shows how far the target x lies between x₀ and x₁ as a percentage. For example, 25% means the point is one quarter of the way from x₀ to x₁.
Can the calculator handle extrapolation?
Yes. If x is outside the interval [x₀, x₁], the same straight-line formula still works and t will fall below 0% or above 100%.
Why must x₀ and x₁ be different?
Because the denominator x₁ - x₀ appears in the formula. If both x-values are equal, the calculation would divide by zero.
Can I use negative or decimal numbers?
Yes. Any finite numbers are allowed as long as x₀ and x₁ are not the same.
Is interpolation always accurate?
It is accurate when the relationship between the two points is approximately linear over the interval. If the data curves strongly, the estimate may be less reliable.
What is the difference between interpolation and extrapolation?
Interpolation estimates within the known interval, while extrapolation estimates beyond it. Extrapolation usually carries more uncertainty.