Calculator
Formula Used
For each polar coordinate, convert to rectangular components.
x₁ = k₁r₁ cos(θ₁)
y₁ = k₁r₁ sin(θ₁)
x₂ = k₂r₂ cos(θ₂)
y₂ = k₂r₂ sin(θ₂)
Add the components.
X = x₁ + x₂
Y = y₁ + y₂
Convert the rectangular sum back to polar form.
R = √(X² + Y²)
θ = atan2(Y, X)
How to Use This Calculator
- Enter the first radius and angle.
- Enter the second radius and angle.
- Use scale factors when adding vector multiples.
- Select the input and output angle units.
- Choose the angle range and decimal precision.
- Press the calculate button.
- Review the polar answer and component table.
- Download CSV or PDF records when needed.
Example Data Table
| r₁ | θ₁ | r₂ | θ₂ | Unit | Result radius | Result angle |
|---|---|---|---|---|---|---|
| 5 | 30 | 3 | 120 | Degrees | 5.8310 | 60.9638° |
| 7 | -40 | 4 | 75 | Degrees | 6.4291 | -5.6756° |
| 10 | 210 | 6 | 330 | Degrees | 8.7178 | -113.4132° |
Understanding Polar Addition
Polar coordinates describe a point by distance and direction. The distance is r. The direction is theta. Adding two polar coordinates is easier after conversion. Each coordinate becomes a horizontal part and a vertical part. Those parts are then added. The final pair is converted back to polar form.
Why Components Matter
A polar pair does not add by adding radii only. Angles control where each radius points. Two long vectors can cancel when they face opposite ways. Two short vectors can build a strong result when they point together. Components protect the calculation from that mistake. They show the real movement along each axis.
Useful Inputs
This calculator accepts degrees, radians, and gradians. It also supports scale values. A scale value lets you add multiples of each vector. This is useful in physics, geometry, navigation, and graphics work. You can also choose the output angle unit. Precision control keeps answers readable for reports.
Reading the Result
The result shows x and y components first. These values are the rectangular form of the sum. The magnitude is the new radius. The angle is the new direction. A quadrant label helps you check the direction quickly. If the magnitude is zero, the angle has no physical direction. The tool reports zero by convention.
Practical Uses
Polar addition appears in vector diagrams. It is used in displacement problems. It helps when combining forces, currents, velocities, and complex numbers. It is also helpful for plotting curves. Surveying and robotics can use the same idea. Any situation with direction and distance can benefit. It also helps check homework. Every component stays visible during review and comparison with manual calculation notes later.
Accuracy Tips
Use one angle unit consistently. Check signs before calculating. Negative radius values are allowed mathematically. They reverse the direction of the coordinate. For clear work, keep radius values positive when possible. Use signed angle output when direction around zero matters. Use positive output when plotting on standard polar grids.
Exporting Work
CSV files are useful for spreadsheets. PDF files are useful for sharing. The export buttons save the inputs, components, formulas, and final polar answer. This makes the calculator useful for class notes, lab reports, and project documentation.
FAQs
What does this calculator add?
It adds two polar coordinates as vectors. Each coordinate is changed into x and y parts. The parts are added, then changed back into polar form.
Can I use radians?
Yes. Select radians as the input unit. You can also choose radians as the output unit for the final angle.
Why not add the two radii directly?
Radii cannot be added alone because angles matter. Two vectors with different directions need component addition for the correct result.
What does the scale factor mean?
The scale factor multiplies each vector before addition. A scale of 2 doubles that coordinate. A scale of -1 reverses its direction.
What is atan2?
atan2 is an angle function that uses both x and y. It places the final angle in the correct quadrant.
Can negative radius values be used?
Yes. Negative radius values are valid in polar math. They act like a direction reversal during component conversion.
What happens when the result radius is zero?
The final vector has no direction. The calculator displays angle zero by convention, but the real direction is undefined.
What export formats are included?
The page includes CSV and PDF downloads. They save inputs, components, selected units, and the final polar result.