Ordered Pair Calculator

Ordered Pair

Convert, reflect, and rotate points easily on the coordinate plane.

How the Ordered Pair Calculator Works (with Formulas)

The Ordered Pair Calculator allows you to perform various mathematical operations on points in the coordinate plane. Below are the operations you can select, together with the formulas used and explanations:

1. Cartesian to Polar Conversion

Input: X and Y

Formulas:

  • Radius (r):
    r = sqrt(x² + y²)

  • Angle (θ, in degrees):
    θ = arctangent(y / x) (expressed as atan2(y, x) for correct quadrant)

Explanation:
Converts Cartesian coordinates (x, y) to polar coordinates (r, θ).

2. Polar to Cartesian Conversion

Input: r (radius), θ (angle in degrees)

Formulas:

  • X coordinate:
    x = r × cos(θ × π / 180)

  • Y coordinate:
    y = r × sin(θ × π / 180)

Explanation:
Converts polar coordinates (r, θ) to Cartesian coordinates (x, y).

3. Find Quadrant

Input: X and Y

Rule:

  • Quadrant I: x > 0, y > 0

  • Quadrant II: x < 0, y > 0

  • Quadrant III: x < 0, y < 0

  • Quadrant IV: x > 0, y < 0

  • If x = 0 or y = 0, the point lies on an axis.

Explanation:
Determines in which quadrant the point (x, y) lies.

4. Reflect over X-axis

Input: X and Y

Formula:

  • Reflected point: (x, -y)

Explanation:
Produces the mirror image of the point across the X-axis.

5. Reflect over Y-axis

Input: X and Y

Formula:

  • Reflected point: (-x, y)

Explanation:
Produces the mirror image of the point across the Y-axis.

6. Reflect over Origin

Input: X and Y

Formula:

  • Reflected point: (-x, -y)

Explanation:
Produces the symmetric point with respect to the origin.

7. Rotate 90° Counterclockwise

Input: X and Y

Formula:

  • Rotated point: (-y, x)

Explanation:
Rotates the point 90 degrees counterclockwise around the origin.

8. Rotate 180°

Input: X and Y

Formula:

  • Rotated point: (-x, -y)

Explanation:
Rotates the point 180 degrees around the origin.

9. Rotate 270° Counterclockwise

Input: X and Y

Formula:

  • Rotated point: (y, -x)

Explanation:
Rotates the point 270 degrees counterclockwise (or 90 degrees clockwise) around the origin.

10. Find Symmetric Point

Input: X and Y

Formula:

  • Symmetric point: (-x, -y)

Explanation:
Finds the point symmetric to (x, y) with respect to the origin.

All calculations are performed instantly after entering the values and selecting an operation.
Use this calculator to understand coordinate geometry, check homework, or visualize point transformations on the plane.