Linear Interpolation Calculator - Proportion-Based y From Two Points

A linear interpolation calculator estimates the missing y at any target x between two known (x, y) points by drawing a straight line through the two points and reading off the y at the target x. The same routine fills gaps in tables, looks up values between known entries, and steps through time-series data one segment at a time. It is the one-dimensional cousin of bilinear interpolation, which extends the same idea to a 2D rectangle.

• • Fill missing data in a table: Pull the y for an x that sits between two tabulated values when the table is monotonic.
• • Look up values in a chart: Estimate the y for a target x that falls between two known axis points when the chart is too coarse to read.
• • Rescale a measurement grid: Convert measurements on a coarse grid to a finer grid using each pair of coarse samples as endpoints of a line.
• • Verify textbook examples: Check the algebra of homework and exam problems that ask for a missing y from two known points and a single proportion.

Linear interpolation is the simplest member of the interpolation family: it assumes the unknown function is a straight line between the two known points.

Use it whenever you have a clean pair of endpoints, the function is roughly linear on the interval, and you only need one value at one target x.

When the two endpoints come from (x, y) coordinates and you also need the geometric distance between them, the 2D Distance Calculator gives the matching line length on the same coordinate plane.

The linear interpolation formula reads off the y at the target x on the straight line through the two known points (x1, y1) and (x2, y2). It computes the slope m = (y2 - y1) / (x2 - x1) once, scales that slope by the horizontal distance (x - x1), and adds the result to y1. The position fraction t = (x - x1) / (x2 - x1) gives the same answer in proportion form: y = y1 * (1 - t) + y2 * t.

• x1, y1: x and y of the first known point. The two endpoints define the line.
• x2, y2: x and y of the second known point. Must have x different from x1 or the slope is undefined.
• x: Target x where the interpolated y is needed. Values outside [x1, x2] extrapolate.
• y: Interpolated y at the target x. Equals y1 at x1, equals y2 at x2, scales linearly in between.

The slope form y = y1 + m * (x - x1) is the spreadsheet-friendly version. The proportion form y = y1 * (1 - t) + y2 * t is the textbook version because it shows y as a weighted average of the two endpoint y-values.

When the two y-values are equal, the slope is 0 and the formula returns that same y for every x. When x equals x1 or x2, the formula returns y1 or y2 exactly.

According to Wikipedia - Linear interpolation, the linear interpolation of a function between two known points (x1, y1) and (x2, y2) at a target x is y = y1 + (x - x1) * (y2 - y1) / (x2 - x1), which is the equation of the straight line through the two endpoints read off at x.

Because the formula is the equation of a line, the Slope Percentage Calculator takes the same (y2 - y1) / (x2 - x1) slope and converts it to a rise-over-run percentage for grade or pitch work.

Four ideas make the result straightforward to read and verify.

These four concepts line up with the calculator's output panel: the slope, the position fraction, and the t-symmetry check are the diagnostic numbers that confirm the algebra.

The same weighted-average view is what makes linear interpolation safe to chain across a long table: each segment has its own slope and its own t, and the result inside that segment is a convex combination of the two endpoint y-values.

Because the position fraction t is just a ratio of the form (x - x1) : (x2 - x1), the Ratio Calculator can reduce the same proportion to its lowest terms and surface the matching scaling factor.

Five quick steps take you from two known (x, y) points to a verified interpolated estimate.

1. 1 Enter the first known point: Type x1 and y1 for the first endpoint. This anchors the left side of the segment.
2. 2 Enter the second known point: Type x2 and y2 for the other endpoint. x2 must differ from x1 or the calculator will show a divide-by-zero error.
3. 3 Enter the target x: Type the x where you want the interpolated y. Values outside [x1, x2] extrapolate and t will fall outside the 0..1 range.
4. 4 Read the result and diagnostics: Check the interpolated y, the slope m, the position fraction t, and the t-symmetry check. The symmetry check is always exactly 1.000000 whenever x2 != x1.
5. 5 Iterate with new endpoints or target: Change any input to update the result on the fly. The position fraction t is the cleanest signal of whether the target is inside the segment.

Try the Wikipedia midpoint example: x1=0, y1=0, x2=10, y2=20, target x=5. The calculator returns y=10 with slope 2, t=0.5, and the t-symmetry check at 1.000000.

If you also want the geometric length of the line between (x1, y1) and (x2, y2) so you can place the interpolation on a sketch, the Length of a Line Segment Calculator returns that hypotenuse length in the same units as your x and y.

Six practical reasons to use this calculator instead of recomputing the proportion by hand.

• • Get the answer in seconds: Skip the handwritten algebra; the calculator applies the closed-form proportion on every keystroke.
• • See the slope and position fraction: The slope m and position fraction t are shown next to y so you can see how the value is built from the two endpoints.
• • Catch divide-by-zero early: Validation stops the calculation when x2 equals x1, so you never see a NaN result from a vertical-line input.
• • Free sanity check with the t-symmetry sum: The t-symmetry check is always exactly 1.000000 whenever x2 != x1, giving a free sanity check that the formula is wired up right.
• • Handle interpolation and extrapolation together: The same calculator returns a clean result when the target x is inside the segment (t in 0..1) and when it is outside (t outside 0..1).
• • Chain segments across a long table: Walk a piecewise-linear path through a table of measurements, with each segment contributing its own slope and t.

These benefits turn the linear interpolation proportion into a one-line operation. Chain the calculator with the related tools to extend the result into slope percentages, line-segment geometry, and best-fit lines.

Watching the position fraction t is the right way to tell whether a target is inside the segment: t in the 0..1 range is an interpolation, t outside that range is an extrapolation.

When the table is too noisy for a single segment and you want a best-fit line through many points instead of a strict pass-through, the Linear Regression Calculator fits that line by least squares and reports the matching slope and intercept.

Five factors determine how reliable the result is.

• • Linear interpolation cannot follow curves. If the true function bends inside the segment, the linear estimate is only an approximation. Switch to a quadratic or cubic interpolant when curvature matters.
• • Linear interpolation is not the same as linear regression. The line is forced to pass through the two endpoints, so any measurement error in the endpoints goes straight into y.

These factors line up with the same limitations that show up in table lookups and time-series analysis. Linear interpolation is fast and stable, but it cannot recover the shape of a curve the way a higher-order interpolant can.

For a single missing data point on a roughly linear segment, linear interpolation is the right tool. For curves, fit a small model first. For noisy data, switch to linear regression.

According to Omni Calculator - Linear Interpolation, linear interpolation estimates the missing y by drawing a straight line between the two known points and reading off the y at the target x, which is the same as scaling the y-difference by the position fraction (x - x1) / (x2 - x1).

When the same straight-line pattern repeats across many evenly spaced x-values and you want the y at index n rather than at one target x, the Arithmetic Sequence Calculator gives the matching arithmetic-sequence term a + (n - 1) * d in one step.