Math
Rotation Calculator
Enter a point, an angle (degrees, radians, or a π fraction), a direction, and optionally a custom center, and mathcheck applies the rotation formula with the 2×2 rotation matrix shown.
For angles that are multiples of 30° or 45° the image is exact, for example ((2 - √2)/2, (2 + 3√2)/2); other angles give four-decimal coordinates. The quick rules for 90°, 180°, and 270° are named when they apply, and the graph shows the point, its image, and the arc between them.
Degrees like 90, radians like 1.5708, or a π fraction like pi/2 (always radians).
Try an example
Result
Rotated point
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- Rotated point (decimal)
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- Rotation matrix
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- Angle used
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- Center
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- Distance from the center
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- Shortcut rule
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
How rotation works
Rotating a point about the origin by θ keeps its distance from the origin and adds θ to its direction angle. Writing that in coordinates gives a pair of formulas involving cos θ and sin θ, which pack neatly into a 2×2 A rectangular grid of numbers arranged in rows and columns.. Rotating about any other center is the same operation after sliding the center to the origin and back.
Shortcut rules for quarter turns
| Counterclockwise angle | Rule | Example (2, 3) → |
|---|---|---|
| 90° | (x, y) → (-y, x) | (-3, 2) |
| 180° | (x, y) → (-x, -y) | (-2, -3) |
| 270° (= 90° clockwise) | (x, y) → (y, -x) | (3, -2) |
| 360° | (x, y) → (x, y) | (2, 3) |
(2, 3) rotated 45° about (1, 1)
Shift: (2 - 1, 3 - 1) = (1, 2). Rotate: x' = 1·(√2/2) - 2·(√2/2) = -√2/2, y' = 1·(√2/2) + 2·(√2/2) = 3√2/2. Shift back: ((2 - √2)/2, (2 + 3√2)/2) ≈ (0.293, 3.121).
How to use this calculator
- Enter the coordinates of the point.
- Type the angle in degrees, An angle unit where a full turn is 2π instead of 360 degrees., or as a π fraction, and choose clockwise or counterclockwise.
- Keep the origin as the center, or choose another point to rotate around.
- Read the image point in exact and decimal form, along with the rotation matrix and any shortcut rule.
- Open Show the work for the trig values, the translation, and each coordinate formula.
How to read the answer
The image is exact when the angle is a multiple of 30° or 45° and the coordinates are decimals; otherwise it is rounded to four places. Counterclockwise is the positive direction in mathematics, so a clockwise request is converted to a negative angle before the matrix is built. The distance from the center is reported as a check: rotation never changes it.
Common mistakes and edge cases
- Using the counterclockwise rule for a clockwise rotation. 90° clockwise is (x, y) → (y, -x), not (-y, x).
- Forgetting to translate when rotating about a point other than the origin.
- Swapping the sign in the matrix: the top-right entry is -sin θ, the bottom-left is +sin θ.
- Typing a radian measure into the degrees box (π/2 is 1.5708 rad, not 1.5708°).
- Rounding cos θ and sin θ before multiplying, which loses the exact A root expression written with the √ symbol, such as a square root..
Worked examples
Quarter turn about the origin
(2, 3) rotated 90° counterclockwise
Rotated point
(-3, 2)
Quarter turn clockwise
(2, 3) rotated 90° clockwise
Rotated point
(3, -2)
Half turn
(-4, 1) rotated 180°
Rotated point
(4, -1)
Exact radicals at 60°
(2, 0) rotated 60° counterclockwise
Rotated point
(1, √3) ≈ (1, 1.7321)
About another center
(2, 3) rotated 45° about (1, 1)
Rotated point
((2 - √2)/2, (2 + 3√2)/2) ≈ (0.2929, 3.1213)
Angle as a π fraction
(1, 1) rotated π/2 radians
Rotated point
(-1, 1)
General angle in decimals
(3, 4) rotated 37°
Rotated point
(-0.0114, 5)
Rotating the center itself
Rotated point
(1, 1)
Unreadable angle
Rotated point
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Frequently asked questions
What is the formula to rotate a point?+
About the origin by θ counterclockwise: x' = x cos θ - y sin θ and y' = x sin θ + y cos θ. About another center, subtract the center first, rotate, then add it back.
How do I rotate a point 90 degrees?+
Counterclockwise: (x, y) → (-y, x). Clockwise: (x, y) → (y, -x). Both come from plugging cos 90° = 0 and sin 90° = ±1 into the general formula.
Is clockwise positive or negative?+
In mathematics counterclockwise is positive. A clockwise rotation by θ is the same as a counterclockwise rotation by -θ, which flips the sign of sin θ in the matrix.
What is the rotation matrix?+
[cos θ -sin θ ; sin θ cos θ]. Multiplying it by the column vector (x, y) gives the rotated point. It has determinant 1, and its inverse is the matrix for -θ.
Why does the calculator give radicals?+
Because at multiples of 30° and 45° the sine and cosine are exact values like √3/2 and √2/2, so the rotated coordinates can be written exactly. Other angles have no such closed form and are shown as decimals.
How do I rotate a whole shape?+
Rotate each vertex with the same angle and center, then reconnect them in the same order. Rotation preserves lengths and angles, so the image is congruent to the original.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026