# What Is Rotational Symmetry?

## Rotational Symmetry

Imagine spinning a basketball like the player in Figure 1. No matter when the spin is stopped, the ball has the same shape. Now imagine rolling dice like in Figure 2. It doesn't matter which face is pointing toward the sky when the dice stop rolling, they will always look the same. These are examples of **rotational symmetry**.

## What is Rotational Symmetry?

The recycling icon is a very common symbol, and like most effective icons, the image itself is suggestive of its meaning. The arrows of the image appear to be moving in a circular manner, suggesting the circular concept of recycling. Adding to this perception is that if you were to rotate the image 120 degrees, and another 120 degrees, and a third 120 degrees, it would look the same at all three stops.

This attribute is called **rotational symmetry**. Many shapes have rotational symmetry, such as rectangles, squares, circles, and all regular polygons. Choose an object and rotate it up to 180 degrees around its center. If at any point the object appears exactly like it did before the rotation, then the object has rotational symmetry. In this lesson, you will be given several image examples, as well as definitions of the relevant concepts **center**, **order**, and **degree of rotation**.

## What Is Rotational Symmetry?

- Rotational Symmetry Definition
- As Figure 1 and Figure 2 illustrate, rotational symmetry involves congruity of a shape during a rotation. Also, for a shape to have rotational symmetry, the shape must look the same before and after a rotation of less than {eq}360^{\circ} {/eq}.

- Determining What Rotational Symmetry Is
- To know if a shape has rotational symmetry, answer the following questionsâ€”if the answer to each bullet point is yes, then the shape in question has rotational symmetry:

- Is the angle of rotation, explained in the following section, less than {eq}360^{\circ} {/eq}?

- If the angle of rotation is less than {eq}360^{\circ} {/eq}, does the after-rotation shape look identical to the before-rotation shape?

### Angle of Rotational Symmetry

The angle of rotational symmetry is how much a shape must be rotated before it looks the same as it did before the rotation. Return to the basketball example of the introduction. Mark a spot on the basketball, and mark a line on the floor. Align the spot and the line, then spin the basketball and mark another line where the spot on the basketball comes to rest. The angle that is formed between these two lines is the angle of rotational symmetry.

The basketball example illustrates continuous rotational symmetry because, no matter how big or how small the angle of rotation is, the basketball will always look the same. The dice of Figure 2, on the other hand, must roll in multiples of {eq}90^{\circ} {/eq} to have rotational symmetry because, only if the dice roll from one face to another, is there symmetry. This is an example of non-continuous rotational symmetry.

### Degree of Rotational Symmetry

The degree of rotational symmetry, also called the order of rotational symmetry, refers to how many times in {eq}360^{\circ} {/eq} a shape can be rotated and display rotational symmetry. For a regular polygon, the order of rotational symmetry is equal to the number of sides of the polygon. Refer again to the basketball whose symmetry is continuous. This sphere can be rotated infinitesimally and always look the same, and all spheres have infinite rotational symmetry. Now think about the dice again. The dice that must be rotated in multiples of {eq}90^{\circ} {/eq} can only be rotated four times before completing a circle. These cubes, like all cubes, have rotational symmetry of order 4.

Another measure of rotational symmetry looks at how many different axes a shape can be rotated around and exhibit rotational symmetry. A basketball has only a single axis, and therefore it can only rotate about that single axis. The dice have six sides, eight vertices, and thirteen possible axes of rotation.

### Rotational Symmetry Shapes

Many shapes have rotational symmetry of at least order 1 with at least one axis of rotation. Figure 3, Figure 4, and Figure 5 illustrate different rotational symmetry shapes. Figure 3 is a tetrahedron which has 24 axes of rotation and a rotational order of 4.

Figure 4 is a 5-sided star, and, because of its five sides, the star has a rotational symmetry of order 5.

Figure 5 is a rhombus. This shape does not follow the regular polygon rule, and a rhombus has a rotational symmetry order of 2 and 2 axes of rotation.

### Rotational Symmetry Graph

Some functions, just like some geometric shapes, exhibit rotational symmetry, and identifying functions that have this symmetry can greatly simplify complex calculus problems. For example, any odd function has rotational symmetry about the origin. Recall that, for an odd function, {eq}f(-x) = -f(x) {/eq}. Figure 6 is the cubic function {eq}f(x) = 5x^3 {/eq}. This function is odd, and, when it is rotated {eq}180^{\circ} {/eq} about the origin, this function exhibits rotational symmetry.

Figure 7 is another rotational symmetry graph. This is the plot of {eq}sin(x) {/eq}, and the rotational symmetry of this function is commonly used by physicists to simplify wave function calculations.

## How to Find Rotational Symmetry

When analyzing a geometric shape, it is useful to know the angle of rotational symmetry and the degree, or order, of rotational symmetry. To find these attributes, use the following set of steps for how to find rotational symmetry:

- Identify if the shape has symmetry before and after rotation. If it does, ensure that the shape has not been rotated 360°.
- Determine the angle of rotation. This can be accomplished by identifying the pivot point of the shape then using trigonometry.
- Once the angle of rotation is known, divide 360° by the angle of rotation. The result is the order of rotational symmetry. For example, for a cube with a 90° angle of rotation, {eq}\frac{360^{\circ}}{90^{\circ}} = 4 {/eq} which is the correct order of rotational symmetry for a cube.

### Rotational Symmetry Examples

Example 1 and Example 2 are both real life rotational symmetry examples. The first rotational symmetry example shows how to find the angle of rotational symmetry and the order of rotational symmetry of an everyday object. The second rotational symmetry example explores how to find the angle of rotational symmetry and the order of rotational symmetry of a flower.

- Example 1
- Maria is ready for a bike ride. As she is heading out of the door she thinks that her helmet is on backwards, but in the hallway mirror it looks normal. Maria takes the helmet off to examine it. Why might Maria have thought her helmet was on backwards?

If Maria's helmet has rotational symmetry of {eq}180^{\circ} {/eq} it is possible that she mistook the back of the helmet for the front of the helmet. To find out if her helmet has rotational symmetry, begin by marking one side of the helmet and a point on the floor, pictured in Figure 8. Spin the helmet about its center of mass which is centralized on the top of the helmet. As the helmet spins a full {eq}360^{\circ} {/eq}, there is one point when it appears to have the same orientation as it did to begin with . Mark this point, pictured in Figure 9. When looking at the angle that is formed between the beginning point and the point when the helmet looks the same as it began with, it can be seen that both lines fall on the y-axis. This means that the angle of rotational symmetry is {eq}180^{\circ} {/eq}. It is no wonder that Maria put her helmet on backwards! Using the angle of rotational symmetry to find the order gives, {eq}\frac{360^{\circ}}{180^{\circ}} = 2 {/eq}. The helmet is oval shaped, and it has an order of rotational symmetry of 2.

- Example 2
- A violet has rotational symmetry because all of its five petals are identical. Find the angle of rotational symmetry and the order of rotational symmetry of a violet.

All of the petals of a violet are identical. Since there are five petals, turning a violet a fifth of a rotation will cause it to look the same as it did before the rotation. This means that the violet can be rotated 5 times in {eq}360^{\circ} {/eq}, giving it a rotational order of 5. The angle of rotational symmetry can be solved be finding x in the expression {eq}\frac{360^{\circ}}{x} = 5 {/eq}:

1) Write the expression.

{eq}\begin{align} \frac{360^{\circ}}{x} = 5 \end{align} {/eq}

2) Multiply both sides of the expression by x and simplify.

{eq}\begin{align} x \times \frac{360^{\circ}}{x} & = x \times 5 \\ 360^{\circ} & = 5x \end{align} {/eq}

3) Isolate x by dividing both sides by 5 and simplifying.

{eq}\begin{align} \frac{360^{\circ}}{5} & = \frac{5x}{5} \\ 72^{\circ} & = x \end{align} {/eq}.

The angle of rotational symmetry of a violet is {eq}72^{\circ} {/eq}.

## Lesson Summary

If an object can be turned and look the same as it did before it was turned, then the object has **rotational symmetry**. Imagine spinning a basketball. No matter where the ball stops it will look the sameâ€”this is rotational symmetry. The angle of rotational symmetry is how many degrees a shape must be rotated before it looks the same as it did before the rotation. The degree of rotational symmetry, also called the order of rotational symmetry, is a measure of how many times a shape can be rotated and exhibit symmetry in a full {eq}360^{\circ} {/eq} turn.

Rotational symmetry has two main attributes:

- the angle of rotational symmetry
- the degree of rotational symmetry

There are many rotational symmetry shapes, e.g., trapezoids, cubes, and stars. There are also rotational symmetry graphs, and many functions exhibit some degree of rotational symmetry. For example, all odd functions have rotational symmetry about the origin. There are many real life examples of rotational symmetry such as in violets, and how to find rotational symmetry is typically done by performing a rotation, then measuring the angle that the shape must turn to look the same as it did before it was turned. Rotational symmetry can be found throughout nature, and it can be used to simplify complex calculus equations in physics and chemistry.

## What is the Center of an Object?

The **center** of a shape or object with rotational symmetry is the point around which the rotation occurs. If one was to spin a basketball on the tip of his finger, the tip of his finger would be the center of the rotational symmetry. If an object has rotational symmetry, its center will also be its center of balance.

## Order of Symmetry

The **order of symmetry** - or for short, order - is the number of times an object or shape can be rotated and still look like it did before rotation began. Let's look at some examples.

Here we have a hexagon. It has an order of 2. Let's look at another example.

Both the blue and orange shape and starfish have an order of 5 because you can turn them 5 times and they still look the same as they did before they were rotated.

Let's look at one more example.

This one has an order of 3.

The smallest order would be an order of 2. You cannot have a shape or object that has an order of 1. An order of 1 would mean that you can complete a full rotation without it appearing as it did before the rotation. In other words, an order of 1 would mean that is has no rotational symmetry.

## What is Degree of Rotation?

The **degree of rotation** is the number of degrees required to rotate the shape or object so that it appears as it did before the rotation. A full rotation requires 360 degrees. To find the degree of rotation, you would divide 360 degrees by the order. Since 2 is the smallest order possible, then 180 is the largest degree of rotation possible.

Let's look back at the hexagon example. It has an order of 2, because it can be turned twice and look the same as it did when we started; so to find out the degree of rotation, we would take 360 (the number of degrees required in a full rotation) and we divide by 2 (the order of symmetry).

360 / 2 = 180

So, the degree of rotation is 180.

Common degrees of rotation are: 180 degrees (order 2); 120 degrees (order 3); 90 degrees (order 4); 72 degrees (order 5); and 60 degrees (order 6). Other, smaller degrees of rotation are possible but rare.

## Lesson Summary

An object has **rotational symmetry** if you can rotate the image around the center and it appears just as it did before the rotation. The number of times that it can be rotated is called the **order of symmetry**. The **degree of rotation** equals 360 degrees divided by the order of rotation, and will range between 0 and 180 degrees.

## Aspects of Rotational Symmetry

Center of an Object | Order of Symmetry | Degree of Rotation |
---|---|---|

The center about which an object is rotated | *The number of times an object can be rotated and still look like the original object * Can't be less than 2 |
*The number of degrees required to rotate the object so that it appears the same as it did before the rotation *360 degrees divided by the order of rotation |

## Learning Outcomes

When you are finished, you should be able to:

- Describe rotational symmetry
- Explain what the order of symmetry and the degree of rotation are and how they are connected
- Recall how to calculate the degree of rotation

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## What is Rotational Symmetry?

The recycling icon is a very common symbol, and like most effective icons, the image itself is suggestive of its meaning. The arrows of the image appear to be moving in a circular manner, suggesting the circular concept of recycling. Adding to this perception is that if you were to rotate the image 120 degrees, and another 120 degrees, and a third 120 degrees, it would look the same at all three stops.

This attribute is called **rotational symmetry**. Many shapes have rotational symmetry, such as rectangles, squares, circles, and all regular polygons. Choose an object and rotate it up to 180 degrees around its center. If at any point the object appears exactly like it did before the rotation, then the object has rotational symmetry. In this lesson, you will be given several image examples, as well as definitions of the relevant concepts **center**, **order**, and **degree of rotation**.

## What is the Center of an Object?

The **center** of a shape or object with rotational symmetry is the point around which the rotation occurs. If one was to spin a basketball on the tip of his finger, the tip of his finger would be the center of the rotational symmetry. If an object has rotational symmetry, its center will also be its center of balance.

## Order of Symmetry

The **order of symmetry** - or for short, order - is the number of times an object or shape can be rotated and still look like it did before rotation began. Let's look at some examples.

Here we have a hexagon. It has an order of 2. Let's look at another example.

Both the blue and orange shape and starfish have an order of 5 because you can turn them 5 times and they still look the same as they did before they were rotated.

Let's look at one more example.

This one has an order of 3.

The smallest order would be an order of 2. You cannot have a shape or object that has an order of 1. An order of 1 would mean that you can complete a full rotation without it appearing as it did before the rotation. In other words, an order of 1 would mean that is has no rotational symmetry.

## What is Degree of Rotation?

The **degree of rotation** is the number of degrees required to rotate the shape or object so that it appears as it did before the rotation. A full rotation requires 360 degrees. To find the degree of rotation, you would divide 360 degrees by the order. Since 2 is the smallest order possible, then 180 is the largest degree of rotation possible.

Let's look back at the hexagon example. It has an order of 2, because it can be turned twice and look the same as it did when we started; so to find out the degree of rotation, we would take 360 (the number of degrees required in a full rotation) and we divide by 2 (the order of symmetry).

360 / 2 = 180

So, the degree of rotation is 180.

Common degrees of rotation are: 180 degrees (order 2); 120 degrees (order 3); 90 degrees (order 4); 72 degrees (order 5); and 60 degrees (order 6). Other, smaller degrees of rotation are possible but rare.

## Lesson Summary

An object has **rotational symmetry** if you can rotate the image around the center and it appears just as it did before the rotation. The number of times that it can be rotated is called the **order of symmetry**. The **degree of rotation** equals 360 degrees divided by the order of rotation, and will range between 0 and 180 degrees.

## Aspects of Rotational Symmetry

Center of an Object | Order of Symmetry | Degree of Rotation |
---|---|---|

The center about which an object is rotated | *The number of times an object can be rotated and still look like the original object * Can't be less than 2 |
*The number of degrees required to rotate the object so that it appears the same as it did before the rotation *360 degrees divided by the order of rotation |

## Learning Outcomes

When you are finished, you should be able to:

- Describe rotational symmetry
- Explain what the order of symmetry and the degree of rotation are and how they are connected
- Recall how to calculate the degree of rotation

To unlock this lesson you must be a Study.com Member.

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- Activities
- FAQs

## Rotational Symmetry and Art

In this creative project, students will create a work of art that has rotational symmetry. Students can use any medium they desire, such as clay, paint, colored pencils, or collage. However, the art that students create should have rotational symmetry and be able to be hung in any direction, as described in the lesson. For example, students might create a mandala like shape from cut outs of colored paper in a color scheme they enjoy. This would be an example of rotational symmetry.

### Student Instructions

Now that you understand what rotational symmetry is and how to identify it, it's time to try your own hand at creating something with this type of symmetry. In this creative assignment you will be creating a work of art that has rotational symmetry. You can choose any medium to work with, such as paint, clay, collage, or something else of your choosing. However, your end product should have rotational symmetry. For a list of constraints, see the criteria for success below.

#### Criteria For Success

- Art should have rotational symmetry
- Art should include a color scheme that is attractive
- The student should be able to discuss their vision for the piece and explain why it has rotational symmetry

#### How do you find the rotational symmetry of a shape?

Rotational symmetry means that a shape or a function can be rotated about a point and look the same as it did before the rotation. To find the rotational symmetry of a shape, first identify if the shape looks the same as it did before the rotation. If it does, make sure the shape was turned less than 360°. If it has been, the shape has rotational symmetry.

#### What is rotational symmetry Order 3?

Rotational symmetry order 3 means that a shape can be rotated 3 times within a full circle and look the same as it did before the rotation. For a shape to be of order 3, it must have an angle of rotation of 120°.

#### What is the rotational symmetry in math?

Rotational symmetry in math means that a shape can be spun around a single point and look the same as it did before it was spun. Many shapes and functions exhibit rotational symmetry.

#### How do you explain rotational symmetry?

Rotational symmetry means that a shape or a function can be spun about a point and look the same as it did before it was spun. Regular polygons and odd functions exhibit rotational symmetry.

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