# Bilinear Interpolation Calculator

URL: https://speedycalculator.com/calculators/math/bilinear-interpolation/
Last updated: 2026-10-30

> The bilinear interpolation calculator estimates a value P at a point (x, y) inside a rectangle from the four values Q₁₁, Q₁₂, Q₂₁ and Q₂₂ at its corners.

## Inputs

- x₁
- x₂
- y₁
- y₂
- Q₁₁ = f(x₁, y₁)
- Q₁₂ = f(x₁, y₂)
- Q₂₁ = f(x₂, y₁)
- Q₂₂ = f(x₂, y₂)
- x
- y

## Results

- Interpolated value P
- Interpolated along y₁ (R₁)
- Interpolated along y₂ (R₂)

## Formula

```
P = [Q₁₁(x₂ − x)(y₂ − y) + Q₂₁(x − x₁)(y₂ − y) + Q₁₂(x₂ − x)(y − y₁) + Q₂₂(x − x₁)(y − y₁)] ÷ [(x₂ − x₁)(y₂ − y₁)]
```

## Explanation

Bilinear interpolation reads values between grid points of a two-way table: steam tables, temperature–pressure charts, digital elevation models and image resizing all use it. It is linear interpolation done first in x and then in y.

### How to use the Bilinear Interpolation Calculator

- **x₁**
- **x₂**
- **y₁**
- **y₂**
- **Q₁₁ = f(x₁, y₁)**
- **Q₁₂ = f(x₁, y₂)**
- **Q₂₁ = f(x₂, y₁)**
- **Q₂₂ = f(x₂, y₂)**
- **x**
- **y**

Results update as you type, and everything is calculated in your browser.

### How it works

The calculator weights each corner value by the area of the opposite sub-rectangle. Points outside the rectangle are extrapolated, which is less reliable.

### Inputs explained

FieldWhat to enterx₁A numberx₂A numbery₁A numbery₂A numberQ₁₁ = f(x₁, y₁)A numberQ₁₂ = f(x₁, y₂)A numberQ₂₁ = f(x₂, y₁)A numberQ₂₂ = f(x₂, y₂)A numberxA numberyA number

### What each result means

- **Interpolated along y₁ (R₁)** = [Q₁₁ = f(x₁, y₁)] + ([Q₂₁ = f(x₂, y₁)] - [Q₁₁ = f(x₁, y₁)])×([x] - [x₁])÷([x₂] - [x₁])
- **Interpolated along y₂ (R₂)** = [Q₁₂ = f(x₁, y₂)] + ([Q₂₂ = f(x₂, y₂)] - [Q₁₂ = f(x₁, y₂)])×([x] - [x₁])÷([x₂] - [x₁])

### Example values

How the main result changes when **x₁** changes and everything else stays at the example values above:

x₁Interpolated value P10141.515137201282510140182

## Worked example

Inputs — x₁: 20; x₂: 30; y₁: 1; y₂: 2; Q₁₁ = f(x₁, y₁): 100; Q₁₂ = f(x₁, y₂): 120; Q₂₁ = f(x₂, y₁): 140; Q₂₂ = f(x₂, y₂): 170; x: 24; y: 1.5.

Results — **Interpolated value P = 128**; Interpolated along y₁ (R₁) = 116; Interpolated along y₂ (R₂) = 140.

## Key terms

- **Interpolation**: Estimating values between known data points.

## FAQ

**What is bilinear interpolation?**
Linear interpolation in two directions inside a rectangle of known values.

**Is the result linear?**
It is linear along lines parallel to the axes but contains an xy term, so overall it is a hyperbolic paraboloid.

**Can I interpolate outside the rectangle?**
Mathematically yes (extrapolation), but accuracy drops.

**Where is it used?**
Lookup tables in engineering, image scaling and GIS.


## Sources

- [Wolfram MathWorld – Bilinear interpolation](https://mathworld.wolfram.com/BilinearInterpolation.html)
- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)