Quick answer
The magnitude of an acceleration vector is |a| = √(aₓ² + a_y² + a_z²). Components of 3 and 4 m/s² give 5 m/s².
Enter the x and y components, and optionally z. The calculator returns the magnitude.
Related tools: Acceleration Calculator: Average acceleration from change in velocity over time. SUVAT Calculator: Solve constant-acceleration motion for the missing quantities. Projectile Motion Calculator: Range, flight time and peak height of a projectile without air resistance.
How to use the Magnitude of Acceleration Calculator
- Enter x component (m/s²).
- Enter y component (m/s²).
- Optionally, enter z component (m/s²).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| x component | A number, in m/s² | — |
| y component | A number, in m/s² | — |
| z component (optional) | A number, in m/s² | — |
Formula and method
Formula
|a| = √(aₓ² + a_y² + a_z²)
IEEE-754 double precision; shown to the decimals indicated.
Vector magnitude.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: x component = 3; y component = 4.
| Result | Value |
|---|---|
| Magnitude | 5 m/s² |
Example 2
Inputs: x component = 1; y component = 2; z component = 2.
| Result | Value |
|---|---|
| Magnitude | 3 m/s² |
Example 3
Inputs: x component = -6; y component = 8; z component = 0.
| Result | Value |
|---|---|
| Magnitude | 10 m/s² |
The first line is the headline result (magnitude). IEEE-754 double precision; shown to the decimals indicated.
Combining components
Perpendicular components combine by Pythagoras: √(9 + 16) = 5. With a z component of 12, the magnitude would be 13. The direction angle in the plane is arctan(4 ÷ 3) = 53.13°.
Use
Resolve a vector into components for analysis, then recombine. Accelerometers in phones report x, y and z components; at rest the magnitude is about 9.81 m/s².
How the result changes with x component
Holding every other input at the first worked example, this table changes only x component. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| x component | Magnitude |
|---|---|
| 1.5 | 4.272002 m/s² |
| 3 | 5 m/s² |
| 6 | 7.211103 m/s² |
| 12 | 12.649111 m/s² |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- x component: left empty shows "Enter x component." beside that field and no result.
- y component: entering "x" shows "y component must be a number." beside that field and no result.
Choosing between the Magnitude of Acceleration Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Acceleration Calculator](/science/acceleration-calculator/) | initial velocity, final velocity and time | Acceleration |
| [SUVAT Calculator](/science/suvat/) | you know and initial velocity u | Final velocity v |
| [Projectile Motion Calculator](/science/projectile-motion-calculator/) | launch speed and launch angle | Range |
| [Free Fall Calculator](/science/free-fall-calculator/) | i know the and height or time | Fall time |
| [Terminal Velocity Calculator](/science/terminal-velocity-calculator/) | mass, frontal area and drag coefficient | Terminal velocity |
Background: motion and mechanics
For constant acceleration the standard relations are v = u + at, s = ut + ½at², v² = u² + 2as and s = ½(u + v)t. Pick the equation that contains your three known quantities and the one you want. Use SI units (metres, seconds, kilograms, newtons, joules) throughout or the answer will be off by a factor.
These equations stop applying when acceleration changes during the motion, as with a rocket burning fuel; that needs calculus or numerical methods.
Assumptions and limitations
- z is optional
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for x component and y component to confirm this.
Frequently asked questions
What is the magnitude if a = (6, 8)?
For a = (6, 8) the magnitude is √(36 + 64) = 10 m/s².
Is magnitude always positive?
Yes. A magnitude is a length, so it is zero or positive; direction is described separately.
Is z required?
No. Leave z blank for two-dimensional motion and the magnitude uses only x and y.
What does the Magnitude of Acceleration Calculator calculate?
Magnitude of an acceleration vector from its components. It returns Magnitude.
What formula does the Magnitude of Acceleration Calculator use?
|a| = √(aₓ² + a_y² + a_z²). Vector magnitude.
What do I need to enter in the magnitude of Acceleration Calculator?
You enter x component (m/s²) and y component (m/s²). Optional: z component (m/s²).
How is the result rounded or limited?
IEEE-754 double precision; shown to the decimals indicated. z is optional.
What is an example of a magnitude of Acceleration result?
With x component 3 and y component 4, the magnitude is 5 m/s².
Related calculators
- Acceleration Calculator — Average acceleration from change in velocity over time.
- SUVAT Calculator — Solve constant-acceleration motion for the missing quantities.
- Projectile Motion Calculator — Range, flight time and peak height of a projectile without air resistance.
- Free Fall Calculator — Fall time, height and impact speed in free fall from rest.
- Terminal Velocity Calculator — Terminal speed of a falling object from mass, area and drag coefficient.
- Angular Acceleration Calculator — Angular acceleration from a change in angular velocity.
Sources
- OpenStax, University Physics Volume 1 (Rice University): mechanics
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.