chevron_left centre of rotation: the point around which a plane shape can rotate and show the same shape in different positions chevron_right

Anna Kowalski

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calendar_month2025-12-04

The Magic Point: Unlocking the Secrets of Rotation

Exploring the centre of rotation, the fixed point that lets shapes spin and still look the same.

Summary: The centre of rotation is a fundamental concept in geometry that describes the fixed point around which a two-dimensional shape turns. When a shape rotates about this special point, it occupies different positions but remains congruent to its original form. Understanding this point is crucial for grasping concepts like rotational symmetry, angle of rotation, and the mechanics behind everyday motions from playground spinners to the hands of a clock. This article will guide you from the basic definition to practical applications, making the invisible point of spin clearly visible.

Defining the Centre of Rotation

Imagine spinning a square piece of paper on the tip of a pencil. The point where the pencil tip meets the paper is the centre of rotation. This is the one point on the plane that does not move during the rotation. Every other point on the shape travels in a circular path around this centre. The distance from the centre to any point on the shape stays constant throughout the turn, which is why the shape's size and shape do not change—only its orientation does.

In more formal terms, a rotation is a rigid transformation. This means it moves a shape without altering its size, shape, or the distances between its points. The two key ingredients for a rotation are:

The centre of rotation (C): The fixed anchor point.
The angle of rotation ($\theta$): How far to turn, measured in degrees or radians (e.g., 90°, 180°).

Formula Tip: If a point $P$ with coordinates $(x, y)$ is rotated about the origin $(0,0)$ by an angle $\theta$, its new position $P'$ is given by: $$P' = (x \cos \theta - y \sin \theta, \; x \sin \theta + y \cos \theta)$$ This formula shows the mathematical magic that keeps distances constant, preserving the shape.

For young learners, think of it as the "bullseye" of the turn. If you've ever spun a fidget spinner, the centre of the bearing is its centre of rotation. The whole toy spins smoothly around that central point.

Finding the Centre: A Step-by-Step Guide

How do you find the centre of rotation if it's not marked? You can discover it by looking at a shape and its rotated image. Here's a simple method using a compass and straightedge (or geometry software):

Pick a point on the original shape (call it $A$) and its corresponding point on the rotated image (call it $A'$).
Draw the perpendicular bisector of the line segment $AA'$. This is a line that cuts $AA'$ in half at a 90° angle.
Pick another pair of corresponding points, $B$ and $B'$.
Draw the perpendicular bisector of the segment $BB'$.
The point where the two perpendicular bisectors intersect is the centre of rotation.

Why does this work? Because during a rotation, every point moves along a circle centred at the centre of rotation. The centre of any circle is always located on the perpendicular bisector of any chord (a line segment connecting two points on the circle). Here, $AA'$ is a chord of the circular path point $A$ travels. Therefore, the centre of that circle—which is our centre of rotation—must lie on its perpendicular bisector.

Rotation vs. Other Transformations

Rotation is one of the four main rigid transformations. It's important not to confuse it with the others, which move shapes in different ways without changing their size.

Transformation	Description	Fixed Points	Simple Example
Translation (Slide)	Moves every point of a shape the same distance in the same direction.	No fixed points (unless moving zero distance).	A puck sliding on ice.
Rotation (Turn)	Turns a shape about a fixed centre point through a given angle.	Only the centre of rotation is fixed.	A windmill blade turning.
Reflection (Flip)	Produces a mirror image of a shape across a line (the line of reflection).	Points lying on the line of reflection are fixed.	A butterfly's wings.
Glide Reflection	A combination of a translation and a reflection parallel to the translation direction.	No fixed points.	Footprints in the sand.

The Link to Rotational Symmetry

A shape has rotational symmetry if it can be rotated less than a full turn (360°) about a point and still look exactly the same as the original. The centre of this turn is called the centre of symmetry or the centre of rotational symmetry.

The order of rotational symmetry tells you how many times the shape matches itself in one full rotation. For example:

An equilateral triangle fits onto itself 3 times during a 360° turn (at 120°, 240°, and 360°). Its order is 3.
A square has an order of 4 (rotations of 90°, 180°, 270°, 360°).
A circle has infinite rotational symmetry, as it looks the same at any tiny angle of rotation.

The centre of rotation for these symmetrical turns is always the geometric centre of the shape. This concept is beautifully illustrated in nature (like starfish and snowflakes) and human-made objects (like logos and wheels).

From Playgrounds to Planets: Real-World Applications

The centre of rotation isn't just a geometry idea; it's a principle that makes our world work. Let's look at some concrete examples where identifying this point is crucial.

1. The See-Saw (Teeter-Totter): The see-saw rotates about a fixed metal bar or fulcrum. This bar is the centre of rotation. When two people sit on either end, they rotate about this central point. The distances from the centre determine how easily one can lift the other (that's the principle of the lever).

2. Clock Hands: The hour, minute, and second hands of an analog clock all rotate around the same centre point—the pin at the middle of the clock face. Their constant rotation at different speeds allows us to tell time.

3. Ferris Wheel: The entire giant wheel rotates around a central axle. This axle is the massive, engineered centre of rotation. Each passenger gondola also rotates around its own smaller centre (a pivot) to stay level, demonstrating a rotation within a rotation.

4. Earth's Rotation: Our planet spins about an imaginary line called its axis. The points where this axis meets the surface are the North and South Poles. The centre of this rotation is actually the centre of the Earth itself. This daily rotation gives us day and night.

5. Sports: When a figure skater pulls their arms in during a spin, they rotate faster but their centre of rotation (roughly along their spine) remains the key point. In a basketball player's spin move, the pivot foot establishes a temporary centre of rotation around which their body turns.

Experiment: Discovering the Centre with Paper

You can become a rotation detective with a simple hands-on experiment.

What you need: A sheet of paper, a pin, a pencil, a ruler, and scissors.

Steps:

Cut out a distinct shape from the paper (like a scalene triangle—one with all sides different).
Place the shape on the table. Place a blank sheet under it.
Stick the pin through a point on your shape that is NOT the centre. Hold the pin firmly—this point will be your chosen centre of rotation.
Rotate the shape a quarter-turn (90°). Trace the shape's new position lightly on the bottom paper.
Rotate it to a half-turn (180°) and trace again. Do the same for a three-quarter turn (270°).
Remove everything. You will see four identical shapes in four different orientations.
Use the perpendicular bisector method described earlier on corresponding points from two different tracings. The lines will intersect at the point where your pin was—the centre of rotation you used!

This experiment visually proves that the centre is the only point common to all positions of the shape.

Question 1: Can the centre of rotation be outside the shape?

Answer: Absolutely. The centre of rotation can be anywhere on the plane: inside the shape, on its edge, or completely outside of it. For example, think of a satellite (the shape) orbiting Earth (a point far outside the satellite). The satellite's path is a rotation around a centre (Earth's centre) that is not part of the satellite itself.

Question 2: What is the difference between the centre of rotation and the centroid (center of mass)?

Answer: They are related but different concepts. The centre of rotation is the chosen fixed point for a specific turning motion. The centroid or center of mass is the unique point where the shape's area or weight is balanced. If you throw a wrench into the air, it will spin naturally around its center of mass. However, you can force it to rotate around any other point (like one end) if you fix that point. In many physics problems, free rotation happens around the center of mass.

Question 3: How is rotation used in technology and design?

Answer: Rotation principles are everywhere in tech. Gears rotate around their central axles to transfer motion and force. Hard drives have spinning platters that rotate at high speed around a central spindle. In computer graphics and animation, every time you turn a character or object in a 3D program, the software calculates rotations around a defined centre point (often called a "pivot point") to render the new position correctly.

Conclusion: The centre of rotation is the silent, fixed anchor of the turning world. From the simple act of opening a door (which rotates on its hinges) to the complex orbits of celestial bodies, this fundamental geometric concept helps us describe, predict, and create motion. Understanding it builds a bridge between basic geometry, art, and physics. By mastering how to locate it and recognizing its role in symmetry and real-life mechanisms, we gain a deeper appreciation for the orderly patterns and motions that surround us every day.

Footnote

¹ Congruent: Having the same shape and size. Two shapes are congruent if one can be transformed into the other by a sequence of rotations, reflections, and translations.
² Rigid Transformation (Isometry): A transformation of a geometric object that preserves the distances between every pair of points, thus preserving the object's size and shape. The main rigid transformations are translation, rotation, and reflection.
³ Perpendicular Bisector: A line that divides a line segment into two equal parts at a 90° angle.
⁴ Chord: A straight line segment whose endpoints both lie on a circle.
⁵ Centroid: The point where the three medians of a triangle intersect; often called the "center of mass" or "balance point" of a two-dimensional shape of uniform density.

#IGCSE #Geometry Transformations #Rotational Symmetry #Angle of Rotation #Rigid Motion #Math in Motion

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