Coordinate Converter Calculator
Cartesian to Polar:
x:
y:
Polar Coordinates:
r = √(x2 + y2)
θ = atan(y/x)
Example: Convert the cartesian coordinates (5, 4) into polar coordinates?
Answer: r = √(52 + 42) = 6.40
θ = atan(4/5) = 38.66
Polar to Cartesian:
r:
θ:
Cartesian Coordinates:
x = r * cos(θ)
y = r * sin(θ)
Example: Convert the polar coordinates (3, 250) into cartesian coordinates?
Answer: x = 3 * cos(250) = -1.03
y = 3 * sin(250) = -2.82
Spherical to Rectangular:
r:
φ:
λ:
Rectangular Coordinates:
x = r * cos(φ) * cos(λ)
y = r * cos(φ) * sin(λ)
z = r * sin(φ)
Example: What are the rectangular coordinates of the spherical coordinates (10, 50, 100)?
Answer: x = r * cos(φ) * cos(λ) = 10 * cos(50) * cos(100) = -1.12
y = r * cos(φ) * sin(λ) = 10 * cos(50) * sin(100) = 6.33
z = r * sin(φ) = 10 * sin(50) = 7.66
Rectangular to Spherical:
x:
y:
z:
Spherical Coordinates:
r = √x2 + y2 + z2
φ = atan(z / √x2 + y2)
λ = atan(y / x) (for x > 0, y > 0)
λ = π + atan(y / x) (for x < 0)
λ = 2π + atan(y / x) (for x > 0, y < 0)
Example: What are the spherical coordinates of the rectangular coordinates (-1.12, 6.33, 7.66)?
Answer: r = √x2 + y2 + z2 = √(-1.12)2+(6.33)2 + (7.66)2 = 10
φ = atan(z / √x2 + y2) = atan(7.66 / √(-1.12)2+(6.33)2) = 50
λ = π + atan(y / x) (for x < 0) = 3.14 + atan(6.33 / -1.12) = 100
Cylindrical to Rectangular:
r:
θ:
z:
Rectangular Coordinates:
x = r * cos(θ)
y = r * sin(θ)
z = z
Example: What are the rectangular coordinates of the cylindrical coordinates (10, 50, 100)?
Answer: x = 10 * cos(50) = 6.43
y = 10 * sin(50) = 7.66
z = z = 100
Rectangular to Cylindrical:
x:
y:
z:
Cylindrical Coordinates:
r = √x2 + y2
θ = cos-1( x / √x2 + y2)
z = z
Example: What are the cylindrical coordinates of the rectangular coordinates (6.43, 7.66, 100)?
Answer: r = √x2 + y2 = √(6.43)2 + (7.66)2 = 10
θ = cos-1( x / √x2 + y2) = cos-1( 6.43 / √(6.43)2 + (7.66)2) = 50
z = z = 100