Interpolation Between Four 4 Points Calculator (with Formula and Examples)

When you’re working with data, sometimes you need to estimate a value that falls between more than two known points. While linear interpolation works great for two points, when you have four known data points, you can get a more accurate estimate using polynomial interpolation or cubic interpolation.

In this article, we’ll explain how interpolation between four points works and give you an interactive calculator you can use right on this page.

🙋‍♀️ What Is Interpolation Between 4 Points?

Interpolation is a method of estimating unknown values that fall between known data points. When you have four known points — say:

  • (x₀, y₀)
  • (x₁, y₁)
  • (x₂, y₂)
  • (x₃, y₃)

—you can use them to fit a cubic polynomial curve and estimate y at any given x within the range of those points.

This technique provides a smoother and more accurate result than connecting just two points with a straight line.

Use the calculator below to interpolate between four known points. Just enter the x and y values for each point, and the x-value you want to estimate.

🔧 4-Point Interpolation Calculator

Enter four known points and the x-value where you want to estimate y.





















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🧮 Example Calculation

Let’s say you know the following four points:

  • (1, 10)
  • (2, 22)
  • (3, 36)
  • (4, 52)

Now you want to find y at x = 2.5. Using the Lagrange formula, you’d compute L₀, L₁, L₂, and L₃ using those x-values, then multiply by each y and add them up.

For a set of Values:

  • (x₀, y₀) = (1, 10)
  • (x₁, y₁) = (2, 22)
  • (x₂, y₂) = (3, 36)
  • (x₃, y₃) = (4, 52)
  • x = 2.5

✅ Step 1: Compute Lagrange basis polynomials at x = 2.5

L₀(2.5):

L₀ = ((2.5 - 2)*(2.5 - 3)*(2.5 - 4)) / ((1 - 2)*(1 - 3)*(1 - 4))
   = (0.5 * -0.5 * -1.5) / (-1 * -2 * -3)
   = (0.375) / (-6)
   = -0.0625

L₁(2.5):

L₁ = ((2.5 - 1)*(2.5 - 3)*(2.5 - 4)) / ((2 - 1)*(2 - 3)*(2 - 4))
   = (1.5 * -0.5 * -1.5) / (1 * -1 * -2)
   = (1.125) / 2
   = 0.5625

L₂(2.5):

L₂ = ((2.5 - 1)*(2.5 - 2)*(2.5 - 4)) / ((3 - 1)*(3 - 2)*(3 - 4))
   = (1.5 * 0.5 * -1.5) / (2 * 1 * -1)
   = (-1.125) / (-2)
   = 0.5625

L₃(2.5):

L₃ = ((2.5 - 1)*(2.5 - 2)*(2.5 - 3)) / ((4 - 1)*(4 - 2)*(4 - 3))
   = (1.5 * 0.5 * -0.5) / (3 * 2 * 1)
   = (-0.375) / 6
   = -0.0625

✅ Step 2: Compute interpolated value

y = L₀*y₀ + L₁*y₁ + L₂*y₂ + L₃*y₃
  = (-0.0625)*10 + (0.5625)*22 + (0.5625)*36 + (-0.0625)*52
  = -0.625 + 12.375 + 20.25 - 3.25
  = 28.75

✅ Final Answer:

The interpolated value of y at x = 2.5 is: 28.75 ✅

It’s a bit time-consuming by hand — which is why we built the calculator above!

🚀 When to Use 4-Point Interpolation

  • Engineering simulations with known sensor values at fixed points
  • Audio waveform smoothing
  • Scientific data fitting
  • Data visualization where smooth curves matter
  • Finance or environmental modeling

📐 Common Method: Cubic Lagrange Interpolation

The most popular method for interpolation using four points is Lagrange Polynomial Interpolation.

Here’s the general idea:

y = L₀(x) * y₀ + L₁(x) * y₁ + L₂(x) * y₂ + L₃(x) * y₃

Each L term is calculated like this:

L₀(x) = ((x – x₁)(x – x₂)(x – x₃)) / ((x₀ – x₁)(x₀ – x₂)(x₀ – x₃))
L₁(x) = ((x – x₀)(x – x₂)(x – x₃)) / ((x₁ – x₀)(x₁ – x₂)(x₁ – x₃))
L₂(x) = ((x – x₀)(x – x₁)(x – x₃)) / ((x₂ – x₀)(x₂ – x₁)(x₂ – x₃))
L₃(x) = ((x – x₀)(x – x₁)(x – x₂)) / ((x₃ – x₀)(x₃ – x₁)(x₃ – x₂))

Plug these into the equation to get the estimated y value.

💡 Final Thoughts

Interpolation between four points gives you a smooth, accurate estimate — especially when data doesn’t follow a straight line. While the math is more complex than linear interpolation, this method is powerful and widely used in technical fields.

Use the calculator to save time and avoid errors in manual calculations.

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