Rotations
A rotation turns a figure around a fixed point by an angle.
About this topic
A rotation turns a figure by a given angle about a fixed point, the centre of rotation. Counterclockwise is positive by convention. About the origin, a 90° rotation sends (x, y) to (−y, x), a 180° rotation sends it to (−x, −y), and 270° sends it to (y, −x).
Rotations preserve size, shape and distances. They describe wheels, clock hands, fans and the spinning of planets, and combine with translations and reflections to describe any rigid motion.
Worked example: rotate the point (2, 3) by 90° counterclockwise about the origin: (x, y) → (−y, x) gives (−3, 2). A 180° turn gives (−2, −3), and 270° gives (3, −2). Rotating clockwise by 90° is the same as 270° counterclockwise. Every point stays the same distance from the centre, so it moves along a circle.
Ferris wheels, steering wheels, windmills and computer screens that turn from portrait to landscape all show rotations. Students often rotate in the wrong direction or mix up the two formulas for 90° and 270°. A reliable method is to sketch the point and check its quadrant after the turn: (2, 3) starts in quadrant I and a quarter turn counterclockwise lands in quadrant II. If the centre is not the origin, shift so the centre is at the origin, rotate, then shift back.
Key ideas
- Common angles: 90°, 180°, 270°
- Counterclockwise is positive
- Distances from the center are preserved
Quick check
Use cases: where Rotations helps
- Homework: review the key idea behind Rotations before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Rotations without notes.
- Visual learning: play with the interactive lab to see how Rotations changes when you adjust its inputs.
- Teaching a friend: explain Rotations in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Rotations?
A rotation turns a figure around a fixed point by an angle.
What will I learn in Rotations?
Common angles: 90°, 180°, 270°. Counterclockwise is positive. Distances from the center are preserved.
Is Rotations part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Rotations lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Rotations?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.