Case 1
- First force F₁
- 30N
- Second force F₂
- 40N
- Angle between them θ
- 90°
50.0000N
Open with these valuesResultant Force Calculator
50.0000N
Result: 50.0000 NTwo forces at an angle combine into one, and its size sits between their difference and their sum. 30 N and 40 N give 70 N pointing the same way, 50 N at a right angle and 10 N head-on. The angle is the one enclosed between the two arrows, in degrees.
Worked examples
Case 2
- First force F₁
- 30N
- Second force F₂
- 40N
- Angle between them θ
- 0°
70.0000N
Open with these valuesCase 3
- First force F₁
- 100N
- Second force F₂
- 50N
- Angle between them θ
- 120°
86.6025N
Open with these valuesHow it's calculated
R = √(F₁² + F₂² + 2 × F₁ × F₂ × cos θ)
- StepEnter both forces in the same unit — newtons are usual.
- StepEnter the angle enclosed between the two arrows, from 0° to 180°.
- ResultRead the magnitude of the single force that replaces them both.
Reference table
| F₁, F₂, angle | What it is | Resultant |
|---|---|---|
| 0, 40, 90 | One force only | 40 |
| 3, 4, 90 | The 3-4-5 right triangle | 5 |
| 5, 5, 180 | Equal and opposite — they cancel | 0 |
| 7, 24, 90 | The 7-24-25 right triangle | 25 |
| 10, 10, 60 | Equal forces 60° apart | 17.3205 |
| 30, 40, 0 | Same direction — they add | 70 |
| 30, 40, 90 | At a right angle | 50 |
| 30, 40, 180 | Opposite — they subtract | 10 |
| 100, 50, 120 | A large and a small force, 120° apart | 86.6025 |
Questions
How do I calculate the resultant of two forces?
Use R = √(F₁² + F₂² + 2 × F₁ × F₂ × cos θ), where θ is the angle enclosed between the two forces. A 30 N and a 40 N force at 90° give √(900 + 1600) = 50 N.
What is a resultant force?
The single force that has the same effect as two or more forces acting together. Instead of tracking each one separately you replace them with one equivalent arrow.
Why is there a plus in front of the cosine?
Because θ is the angle between the two force arrows, not the angle inside the triangle they form. Those two angles are supplementary, and swapping one for the other flips the sign — with the triangle angle the same formula carries a minus.
What happens at 0° and at 180°?
At 0° the cosine is 1 and the forces add: R = F₁ + F₂, the largest result possible. At 180° the cosine is −1 and they subtract: R = |F₁ − F₂|, the smallest. Everything else lies between those two.
Degrees or radians?
Degrees, from 0 to 180, and the calculator converts internally. Reading 60 as radians instead of degrees would turn 17.32 N into 4.4 N, so the difference is not cosmetic.
Sources and last check
Information, not professional advice.