Coordinate systems (and when to use each)

A coordinate system is just a way to describe the same point in space using different numbers. The “best” system is usually the one that matches the symmetry of your problem. A circle around the origin is easier in polar coordinates; a cylinder is easier in cylindrical coordinates; and many 3D distance problems become simpler in spherical coordinates.

• Cartesian (x, y, z): distances along perpendicular axes. Great for grids, linear algebra, and most graphs.
• Polar (r, θ): radius + angle in 2D. Great for circles, spirals, rotations, and “distance + direction”.
• Cylindrical (r, θ, z): polar in the x–y plane plus height z. Great for cylinders and pipes.
• Spherical (ρ, θ, φ): distance from origin plus two angles. Great for spheres, cones, and many physics fields.

• θ (theta): azimuth angle in the x–y plane measured from the +x axis toward +y.
• φ (phi): polar angle measured down from the +z axis (0° at the “north pole”, 90° in the x–y plane).

Frequently Asked Questions

• What’s the difference between polar and cylindrical?

  Polar is a 2D system: (r, θ). Cylindrical extends polar into 3D by adding height: (r, θ, z). Same idea — just one more coordinate.

• Why does θ sometimes come out negative?

  Angles can be represented in multiple equivalent ways. For example, −30° is the same direction as 330°. This calculator uses atan2(y, x) when appropriate to keep the quadrant correct. You can always add 360° (or 2π) to make a negative angle positive.

• Degrees or radians — which should I use?

  Use whatever your class or tool expects. Most hand-calculation homework uses degrees, while calculus and programming often use radians. This page lets you pick either.

• Some textbooks define φ differently. Which one is “right”?

  Both are common. Here, φ is measured from the +z axis. Other texts use φ as elevation above the x–y plane. If your φ is elevation (call it φ_elev), then φ (this page) = 90° − φ_elev (or π/2 − φ_elev).

• Can I use this to check homework?

  Yes — it’s great for verifying conversions. Still, write the steps in your own work, especially for showing how you handled the angle quadrant and your chosen convention.

Cartesian coordinates converter: formulas, examples, and practical tips

If you’ve ever drawn a point on graph paper, you’ve used Cartesian coordinates. A point in 2D is written as (x, y): move x units left/right and y units up/down. In 3D, you add a third axis, (x, y, z), to represent depth/height. Cartesian coordinates are perfect for grids, straight lines, and matrix-style math — but they can feel clunky when your shape has natural “round” symmetry.

That’s where polar, cylindrical, and spherical coordinates shine. Instead of describing a point by how far it is along an axis, these systems describe a point by how far it is from an origin and what direction it points. This often turns long algebra into short, clean formulas. For example: circles and spirals are easier in polar; pipes and cylinders are easier in cylindrical; and anything with a “ball-like” symmetry becomes simpler in spherical.

In 2D, polar coordinates represent the same point as (r, θ). Think of r as the distance from the origin, and θ as the angle from the +x axis toward +y. The key triangle is formed by dropping a perpendicular to the x axis. From that right triangle:

• r = √(x² + y²)
• θ = atan2(y, x) (quadrant-correct angle)
• x = r cos θ
• y = r sin θ

The important detail is atan2(y, x). A plain arctan(y/x) can’t tell whether the point is in Quadrant II or IV because the ratio y/x can look the same with different signs. The atan2 function uses the signs of both x and y to return the correct quadrant automatically. If you’re doing it by hand, you can mimic atan2 by computing arctan(|y/x|) and then adjusting based on the quadrant.

Suppose your point is (x, y) = (3, 4). Then: r = √(3² + 4²) = √(9 + 16) = 5. θ = atan2(4, 3) ≈ 53.1301° (or 0.9273 rad). So the polar form is (r, θ) ≈ (5, 53.13°). Going back: x = 5 cos(53.13°) ≈ 3, and y = 5 sin(53.13°) ≈ 4.

Cylindrical coordinates extend polar coordinates into 3D: (r, θ, z). The (r, θ) part describes where you are in the x–y plane, and z stays the same as Cartesian z. The conversion is basically “do polar in the base plane, keep height unchanged”:

• r = √(x² + y²)
• θ = atan2(y, x)
• z = z
• x = r cos θ
• y = r sin θ
• z = z

Cylindrical coordinates are popular in engineering because so many real-world objects are cylindrical: pipes, motors, axles, tanks, and circular motion problems. If you’re integrating over a cylinder in calculus, switching to cylindrical coordinates can turn a messy region into a clean set of bounds.

Take (x, y, z) = (−2, 2, 7). Then r = √(4 + 4) = √8 ≈ 2.828. θ = atan2(2, −2) = 135° (because the point is in Quadrant II). So cylindrical is (r, θ, z) ≈ (2.828, 135°, 7). Converting back: x = 2.828 cos(135°) ≈ −2 and y = 2.828 sin(135°) ≈ 2.

Spherical coordinates describe a point in 3D using (ρ, θ, φ). Here ρ is the distance from the origin to the point. θ is the same azimuth angle as before in the x–y plane. φ is the polar angle measured down from the +z axis (0° at the top, 90° in the x–y plane). With this convention:

• ρ = √(x² + y² + z²)
• θ = atan2(y, x)
• φ = arccos(z / ρ) (when ρ > 0)
• x = ρ sin φ cos θ
• y = ρ sin φ sin θ
• z = ρ cos φ

Why arccos(z/ρ)? Because z/ρ is the cosine of the angle between the point and the +z axis. If you prefer thinking in terms of “elevation” above the x–y plane, note that elevation = 90° − φ. Again: different textbooks swap symbols, so always confirm your course convention.

Consider (x, y, z) = (1, 2, 2). Then ρ = √(1 + 4 + 4) = √9 = 3. θ = atan2(2, 1) ≈ 63.4349°. φ = arccos(2/3) ≈ 48.1897°. So spherical is (ρ, θ, φ) ≈ (3, 63.43°, 48.19°). Convert back: x = 3 sin(48.19°) cos(63.43°) ≈ 1, y ≈ 2, z ≈ 2.

• Quadrants: Use atan2(y, x) (or adjust manually). Don’t rely on arctan(y/x) alone.
• Angle units: Degrees vs radians. If you plug degrees into a calculator expecting radians, results will be wildly off.
• Negative radius: In polar/cylindrical, r is usually nonnegative. If you get a negative r, you can flip the direction by adding 180° (or π) to θ.
• Spherical φ convention: This page uses φ from +z. If your text uses elevation, convert with φ = 90° − elevation.
• Rounding: Small rounding differences are normal. If you need exact forms, keep radicals and symbolic trig where possible.

Converting coordinates isn’t just a “homework step.” It’s often the smartest way to simplify a problem: polar makes rotating vectors easy, cylindrical makes circular symmetry easy, and spherical often makes 3D distances and radially symmetric integrals dramatically simpler. In physics, spherical coordinates show up in gravitational and electric fields; in robotics, polar conversions help with motion planning and sensor interpretation; and in computer graphics, spherical angles are everywhere for cameras and lighting.

If you’re building a mental shortcut: Cartesian is “grid thinking,” polar is “distance + direction in a plane,” cylindrical is “polar + height,” and spherical is “distance + direction in 3D.” Once that clicks, the formulas above start to feel inevitable rather than memorized.