Math July 13, 2026 · 8 Min Read

Center of Mass Calculator – Guide & Formulas

Calculate the center of mass (centroid) of a system of point masses in 2D or 3D. Enter mass and position values for instant results.

Try the free calculator

Put these formulas into practice with our instant, step-by-step Center of Mass Calculator.

Open Calculator ›

The Center of Mass Calculator finds the balance point of a system of point masses in 2D or 3D space. Enter the mass and position coordinates of each object, and the calculator determines the exact center of mass using the weighted average formula.

Key Takeaway

Use the free Center of Mass Calculator to calculate the center of mass (centroid) of a system of point masses in 2d or 3d. enter mass and position values for instant results. Get instant results with step-by-step explanations.

How to Use the Center of Mass Calculator

  1. Select 2D or 3D mode depending on your problem.
  2. Enter the mass and position coordinates (x, y, and optionally z) for each object.
  3. Add or remove objects as needed using the buttons.
  4. Review the calculated center of mass coordinates and total system mass.

The Formula

X_cm = Σ(m_i × x_i) / Σ(m_i) | Y_cm = Σ(m_i × y_i) / Σ(m_i) | Z_cm = Σ(m_i × z_i) / Σ(m_i)

Variable Definitions

  • x_cm: X-coordinate of the center of mass
  • y_cm: Y-coordinate of the center of mass
  • m_i: Mass of the i-th object in the system
  • x_i: X-position of the i-th object
  • y_i: Y-position of the i-th object

Example: Three Masses in 2D

Find the center of mass of three objects: m₁=2 kg at (1,3), m₂=3 kg at (4,1), m₃=5 kg at (2,5).

  1. Step 1: Calculate the total mass: M = 2 + 3 + 5 = 10 kg.
  2. Step 2: Calculate Σ(m_i × x_i): (2×1) + (3×4) + (5×2) = 2 + 12 + 10 = 24.
  3. Step 3: Calculate Σ(m_i × y_i): (2×3) + (3×1) + (5×5) = 6 + 3 + 25 = 34.
  4. Step 4: X_cm = 24/10 = 2.4, Y_cm = 34/10 = 3.4.
  5. Step 5: The center of mass is at (2.4, 3.4).

Frequently Asked Questions