Calculator
Formula used
The calculator uses trigonometric projection. It projects the radius onto each perpendicular axis.
Polar 2D: x = r cos(θ), y = r sin(θ)
Cylindrical 3D: x = ρ cos(θ), y = ρ sin(θ), z = z
Spherical polar: x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), z = ρ cos(φ)
Spherical elevation: x = r cos(e) cos(a), y = r cos(e) sin(a), z = r sin(e)
All angles are converted to radians internally. Then sine and cosine are applied.
How to use this calculator
- Select the coordinate system you want to convert.
- Enter the radius or distance value.
- Enter the azimuth angle or θ value.
- Add z, φ, or elevation when the selected mode needs it.
- Choose the correct angle unit.
- Set decimal places and press the calculate button.
- Review the rectangular point, checks, and formula note.
Example data table
| Source type | Input | Formula focus | Expected rectangular output |
|---|---|---|---|
| Polar | r = 10, θ = 30° | x = r cos(θ), y = r sin(θ) | (8.660254, 5.000000) |
| Cylindrical | ρ = 8, θ = 135°, z = 12 | Keep z unchanged. | (-5.656854, 5.656854, 12.000000) |
| Spherical polar | ρ = 20, θ = 45°, φ = 60° | φ is measured from +z. | (12.247449, 12.247449, 10.000000) |
Understanding Rectangular Coordinate Conversion
Rectangular coordinates describe position with perpendicular axes. In two dimensions, the point uses x and y. In three dimensions, it also uses z. Many measurements arrive in another coordinate form. Polar data may describe distance and direction. Cylindrical data may describe radius, angle, and height. Spherical data may describe distance with two angles. This calculator converts those values into rectangular coordinates. It also keeps the formula visible, so each result stays easy to check.
Why Rectangular Coordinates Matter
Rectangular coordinates are common in graphs, maps, engineering drawings, computer models, and physics problems. They are often easier to add, subtract, plot, and compare. A polar point can show where an object is from an origin. A rectangular point shows how far it moves along each axis. That difference is useful when plotting vectors, locating points, or combining several movements. Rectangular form also works well with spreadsheets, design software, and programming tasks.
Common Source Systems
Polar coordinates use a radius and one angle. They work well for flat circles, bearings, and rotating motion. Cylindrical coordinates extend polar form by adding height. This is useful for pipes, tanks, antennas, and circular structures. Spherical coordinates use a radius and two angles. They are useful for navigation, astronomy, robotics, lighting, and three dimensional geometry. The calculator supports both common spherical conventions. One uses a polar angle from the positive z axis. The other uses elevation from the horizontal plane.
Precision and Angle Units
Correct angle units are very important. Degrees are common in school work and field measurements. Radians are common in advanced math and many programming environments. Gradians and turns appear in surveying or special workflows. The calculator converts the selected unit before applying sine and cosine. You can also choose the number of decimal places. Higher precision is helpful for design checks. Lower precision is easier to read in reports.
Practical Notes for Reliable Results
Use consistent units for radius, height, and coordinate labels. If the radius is negative, the direction may flip compared with a positive radius. For most practical work, use a nonnegative radius and let the angle describe direction. Check whether your spherical source defines the second angle as polar angle or elevation. These two conventions produce different z values. Review the output vector, table row, and formula notes together. They make the conversion transparent and reduce mistakes.
Example Uses
A surveyor may convert a distance and bearing into east and north offsets. A designer may convert a cylindrical point into machine coordinates. A student may check a spherical coordinate answer before graphing it. A developer may need x, y, and z values for a canvas, simulation, or game engine. In every case, rectangular output gives direct axis movement. That makes the point easier to store, display, and reuse. It also supports quick validation when results must match drawings, tables, formulas, or known reference points closely.
Frequently Asked Questions
What are rectangular coordinates?
Rectangular coordinates describe a point with perpendicular axis values. In 2D, they use x and y. In 3D, they use x, y, and z.
Can this calculator convert polar coordinates?
Yes. Select the polar option. Enter radius and angle. The tool returns x and y using cosine and sine projections.
Can it convert cylindrical coordinates?
Yes. Select cylindrical mode. Enter radius, angle, and z. The x and y values are projected, while z stays unchanged.
Which spherical convention should I choose?
Choose spherical polar when φ is measured from the positive z-axis. Choose spherical elevation when the second angle is measured above or below the horizontal plane.
Why do angle units matter?
Sine and cosine need a known angle unit. A value of 90 degrees is not the same as 90 radians. Select the unit used by your source data.
What does angle normalization do?
Normalization changes angles to an equivalent value from 0 to one full turn. It can make very large or negative angles easier to compare.
Can radius be negative?
The tool can process a negative radius. However, most real measurements use a nonnegative radius and let the angle define direction.
How many decimals can I show?
You can choose from 0 to 10 decimal places. More decimals help with technical checks. Fewer decimals make reports easier to read.
Why is z blank for polar mode?
Polar coordinates are two dimensional. They convert to x and y only. Use cylindrical or spherical mode when you need z.
What is the xy radius check?
The xy radius check equals the square root of x squared plus y squared. It helps confirm the horizontal projection of the point.
Can I use custom unit labels?
Yes. Add a label such as meters, feet, or pixels. The label appears beside the output values without changing the calculation.