Interpolation Calculator: Linear & Lagrange Polynomial

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Formula at a glance

Interpolation Calculator: Linear interpolation — y = y₁ + (x − x₁)·(y₂ − y₁)/(x₂ − x₁). Example: Between (2, 4) and (3, 9) → Interpolated y at x = 2.5: 6.5 — How the Interpolation Calculator works: Linear interpolation — y = y₁ + (x − x₁)·(y₂ − y₁)/(x₂ − x₁)

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📘 About the Interpolation Calculator

Interpolation estimates an unknown value that lies between known data points. Linear interpolation joins two neighbouring points with a straight line — the everyday method for reading statistics tables, steam tables, tax slabs and survey data. Lagrange interpolation fits one smooth polynomial through all the points.

Enter two points and the x you want, or paste a list of points. The calculator shows the linear formula with your numbers, the Lagrange polynomial through all the points (with its equation when the x values are whole numbers), warns when you are extrapolating outside the data, and graphs both.

🧮 Formula

Linear interpolation

y = y₁ + (x − x₁)·(y₂ − y₁)/(x₂ − x₁)

Lagrange polynomial

P(x) = Σ yᵢ·Π (x − xⱼ)/(xᵢ − xⱼ)

✏️ Worked examples

Between (2, 4) and (3, 9)

You enter
x₁	2
y₁	4
x₂	3
y₂	9
Find y at x =	2.5
Method	Linear (straight line between neighbours)

You get
Interpolated y at x = 2.5	6.5
Linear interpolation	y = y₁ + (x − x₁)(y₂ − y₁)/(x₂ − x₁) = 4 + (2.5 − 2)(9 − 4)/(3 − 2) = 6.5
Slope between the two points	5

y = 6.5.

Through 4 points on y = x²

You enter
x₁	2
y₁	4
x₂	3
y₂	9
Find y at x =	2.5
Or many points (optional)	1,1; 2,4; 3,9; 4,16
Method	Polynomial through all points (Lagrange)

You get
Interpolated y at x = 2.5	6.25
Linear interpolation	y = y₁ + (x − x₁)(y₂ − y₁)/(x₂ − x₁) = 4 + (2.5 − 2)(9 − 4)/(3 − 2) = 6.5
Polynomial (Lagrange) through all 4 points	6.25 — y = x²
Slope between the two points	5

y = 6.25.

🧭 How to use this calculator

What you enter

x₁
y₁
x₂
y₂
Find y at x =
Or many points (optional)Overrides x₁…y₂.
Method

What you get

Interpolated y at x = 2.5
Linear interpolation
Slope between the two points

❓ Frequently asked questions

What is linear interpolation?

Assuming a straight line between two known points: halfway between x₁ and x₂, y is halfway between y₁ and y₂.

What is the difference between interpolation and extrapolation?

Interpolation estimates inside the range of the data; extrapolation goes beyond it and is much less reliable.

When should I use polynomial interpolation?

For smooth data with a few points. With many points high-degree polynomials can wiggle — use linear or spline interpolation instead.

📍 Interpolation Calculator (Linear & Lagrange)