# An Overview of Point Symmetry

## What is Point Symmetry?

**Point symmetry** is when, given a central point on a shape or object, every point on the opposite sides is the same distance from the central point. Other terms for point symmetry include origin symmetry (origin is another word for the central point) and rotational symmetry. When viewed from opposite directions, the opposite sides or parts will look the same. To test if an object has point symmetry, rotate it on its central point or origin. If it has point symmetry, when rotated, it will match up with the other side.

In this rectangle, the central point or origin can be seen, labeled O. The shape has two diagonals demonstrating that the vertices are all equal distance from the origin.

## What Is Symmetry?

Before we explore the definition of symmetry, let's complete an activity. Draw an uppercase X on a piece of paper. At the point where the lines cross, place a noticeable dot or point. What do you notice? Do you see two Vs, but one is upside down?

Now, draw an S. Is there a place on the S where you could place a point so that you have the same effect as with the X? If you chose right in the center of the S, then you are correct.

**Point symmetry** occurs when there exists a position or a central point on an object such that:

- The central point splits the object or shape into two parts.
- Every part on each has a matching part on the other that is the same distance from the central point.
- Both parts face different directions.

Let's test our definition with the X and S. Notice the point splits both letters into two similar shapes, but they face different directions.

If you walk up to a mirror and touch the mirror with your finger, you would have made an example of point symmetry. Right where your finger touches the mirror is the point. It's as if you're connected to your image. That is the most important concept of point symmetry: there has to be a connection.

## When Does Point Symmetry Occur?

Point symmetry occurs in a variety of shapes and objects. In order for point symmetry to occur:

- There must be a central point that divides the shape or object into two sides or parts, here referred to as Part A and Part B.
- Every point on Part A must have a matching point on Part B that is the same distance away from the central point.
- Part A and Part B are facing opposite directions.

## Point Symmetry vs. Reflection

**Line symmetry**, or **reflection**, is when an object or shape has a line of symmetry that, if folded in half on this line, one side would match the other perfectly. Think of a person looking at their reflection in a pond. This is not the same as point symmetry. It is possible for a shape or object to have line symmetry, but not point symmetry. It is also possible for a shape or object to have both line symmetry and point symmetry. A shape can also have multiple lines of symmetry. The following are examples of line symmetry or reflection:

#### Line Symmetry Example One

This triangle has line symmetry because a line of symmetry can be drawn from one vertex to the midpoint of the opposite side. This line of symmetry would divide the shape perfectly in half. If folded along the line of symmetry, one side would match the other side perfectly. The triangle has three total lines of symmetry. It does not have point symmetry. If a central point was drawn in the middle, the points on one side would not all be the same distance from the central point as the points on the other side.

#### Line Symmetry Example Two

This star has line symmetry because if a line of symmetry is drawn down the center and folded in half on the line of symmetry, one side would match the other. It has five total lines of symmetry. It does not have point symmetry. If a central point was drawn, the points on one side would not be equal distance from the central point as the points on the other side.

#### Line Symmetry Example Three

This circle has line symmetry and point symmetry. If folded in half, both sides would match up. If rotated 180 degrees, the image would be the same.

## Point Symmetry Examples

#### *Point Symmetry Example One

A playing card has point symmetry. If a dot is drawn in the center of the playing card, that would serve as a central point, dividing the card into opposite sides facing opposite directions. All of the points on one side would match the points on the other side. If rotated 180 degrees, the sides would match up. This is point symmetry.

#### *Point Symmetry Example Two

The letter H has point symmetry. If a dot is drawn in the center of the letter H, that would create a central point. All of the points on one side of the central point match the points on the other side. If rotated 180 degrees, one would see the same figure. This is point symmetry. Many other letters are also examples of point symmetry such as X, N, and Z.

#### *Point Symmetry Example Three

This cathedral's ceiling is an example of point symmetry. There is a visible central point in the middle of the shape. All the points on one side of the central point are the same distance from the

central point as the points on the other side. If rotated 90 or 180 degrees, one would see the same figure. This is point symmetry.

## Lesson Summary

**Point symmetry** is when a shape or object meets the following conditions:

- A central point can be drawn that divides it into two sides.
- All of the points on one side are the same distance from the central point as all of the points on the other side.
- The two sides are facing opposite directions.

When looking at an object or shape with point symmetry from the opposite side, it will look the same. Rotate a shape and see if it matches up with the opposite side to test if it has point symmetry. **Reflection**, or **line symmetry** is often confused with point symmetry. Line symmetry is when a line of symmetry can be drawn through an object or shape and one side matches the other side perfectly. If a shape were to be folded along its line of symmetry, the two opposite sides would align or match up. It is possible for a shape or object to have both point and line symmetry. It is also possible for an object to have line symmetry, but not point symmetry.

## Point Symmetry vs. Reflection

There is usually a misconception about symmetry and reflection. The difference is in the connection. It's as if you are connected to the image. If you stand three feet away from the mirror, you and your image are separate entities. However, with point symmetry, the object is not away from the image, it's onto the image.

Notice that we didn't separate the X or S; all we did was add the point to show that these letters have point symmetry. There are a few more uppercase letters in the alphabet that have point symmetry; try to discover them. Just ensure that they satisfy each criteria we talked about before.

## Examples of Point Symmetry

We already explored the S and the X. So, let's look at another example.

As you can see, an hour glass has point symmetry.

The letter Q has no point symmetry. Although this letter looks like a circle and may seem to have symmetry, it doesn't because of the extended line or tail on the Q, which will only be in one corner. Both halves will not be the same.

Another familiar example of an object with point symmetry is a square. Some cards in a regular deck also have point symmetry. You could try to observe items in your home that have point symmetry - maybe your ceiling fan or even your fancy dishes . Many flowers in nature also have point symmetry.

## Lesson Summary

An object or shape has **point symmetry** if, by inserting a point on the object or shape, two similar shapes are formed, but they face different directions. It's like you are splitting the object in half so that both sides are the same but facing a different direction. Every corresponding point on each shape is the same distance away from the central point. In point symmetry, there has to be a central point that connects the shapes.

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

Create your account

## What Is Symmetry?

Before we explore the definition of symmetry, let's complete an activity. Draw an uppercase X on a piece of paper. At the point where the lines cross, place a noticeable dot or point. What do you notice? Do you see two Vs, but one is upside down?

Now, draw an S. Is there a place on the S where you could place a point so that you have the same effect as with the X? If you chose right in the center of the S, then you are correct.

**Point symmetry** occurs when there exists a position or a central point on an object such that:

- The central point splits the object or shape into two parts.
- Every part on each has a matching part on the other that is the same distance from the central point.
- Both parts face different directions.

Let's test our definition with the X and S. Notice the point splits both letters into two similar shapes, but they face different directions.

If you walk up to a mirror and touch the mirror with your finger, you would have made an example of point symmetry. Right where your finger touches the mirror is the point. It's as if you're connected to your image. That is the most important concept of point symmetry: there has to be a connection.

## Point Symmetry vs. Reflection

There is usually a misconception about symmetry and reflection. The difference is in the connection. It's as if you are connected to the image. If you stand three feet away from the mirror, you and your image are separate entities. However, with point symmetry, the object is not away from the image, it's onto the image.

Notice that we didn't separate the X or S; all we did was add the point to show that these letters have point symmetry. There are a few more uppercase letters in the alphabet that have point symmetry; try to discover them. Just ensure that they satisfy each criteria we talked about before.

## Examples of Point Symmetry

We already explored the S and the X. So, let's look at another example.

As you can see, an hour glass has point symmetry.

The letter Q has no point symmetry. Although this letter looks like a circle and may seem to have symmetry, it doesn't because of the extended line or tail on the Q, which will only be in one corner. Both halves will not be the same.

Another familiar example of an object with point symmetry is a square. Some cards in a regular deck also have point symmetry. You could try to observe items in your home that have point symmetry - maybe your ceiling fan or even your fancy dishes . Many flowers in nature also have point symmetry.

## Lesson Summary

An object or shape has **point symmetry** if, by inserting a point on the object or shape, two similar shapes are formed, but they face different directions. It's like you are splitting the object in half so that both sides are the same but facing a different direction. Every corresponding point on each shape is the same distance away from the central point. In point symmetry, there has to be a central point that connects the shapes.

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

Create your account

#### What is meant by point symmetry?

Point symmetry is when a shape or object has a central point that divides it into two sides. All points on one side are the same distance from the central point as the points on the other side.

#### What shapes have a point symmetry?

Any shape that has a central point that divides it into two sides, with all points on one side the same distance from all points on the other side. Some common examples are rectangles, squares, and circles.

### Register to view this lesson

### Unlock Your Education

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.

Become a MemberAlready a member? Log In

Back### Resources created by teachers for teachers

I would definitely recommend Study.com to my colleagues. Itâ€™s like

**a teacher waved a magic wand and did the work for me.** I feel like itâ€™s a lifeline.