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Introduction

QUICK LINKS


Introduction

Rotations

A rotation is a type of rigid transformation, which means it changes the position or orientation of an image without changing its size or shape.

A rotation does this by rotating an image a certain amount of degrees either clockwise ↻ or counterclockwise ↺.
For rotations of , , and in either direction around the origin , there are formulas we can use to figure out the new points of an image after it has been rotated.
Clockwise ↻Counterclockwise ↺
When we simplify, we can see that the counterclockwise and clockwise rotations are just the reverse of each other. So there are really only 3 rotation formulas we need to remember.
To help us understand and remember the rotation formulas, we like to imagine two arrows rotating.
The arrows start pointing in the positive and directions. As they rotate, we use their new positions to determine the changes in and .
Remember, there are in a circle, which means that each quarter turn is .
Try rotating different points around the origin:
Point:Direction:Angle of Rotation:
Check out our step-by-step rotation calculator to practice rotating images or keep scrolling to learn how to rotate images around the origin and other points.

Rotation Calculator

Step 1. Identify the center of rotation.

Step 2. Identify the original points.

How many points does your original have?
Enter the original points:
Original Points

Step 3. Subtract the center of rotation from each original point.

We're basically translating the shape to the origin so that we can use our rotation formulas.

Step 3. Identify the angle and direction of the rotation.

From the question, we know we are rotating the shape .
Direction:Angle of Rotation:

Step 4. Identify the formula that matches the rotation.

Key Steps 🗝 How to Perform Rotations

Step 1. Identify the center of rotation.

Step 2. Identify the original points.

Step 3. Identify the angle and direction of rotation.

Direction:Angle of Rotation:

Step 4. Identify the formula that matches the rotation.

When we rotate counterclockwise:
  • The axis lines up with the arrow pointing in the negative direction, so the new value is the negative of the old value.
  • The axis lines up with the arrow pointing in the positive direction, so the new value is the old value.

Step 5. Apply the formula to each original point to get the new points.

Original PointNew Point
......

Step 6. Plot the new points.

Continue to explore rotations through some example problems!

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Rotations around the Origin

We can use the following formulas for clockwise and counterclockwise rotations of , , and :
RotationFormula
clockwise or counterclockwise
clockwise or counterclockwise
clockwise or counterclockwise
Remember clockwise (↻) means the direction in which a clock’s hands would normally rotate, and counterclockwise (↺) means the opposite of that.

Clockwise ↻

Counterclockwise ↺

Once we know which formula we need to use, we can apply the formula to the and values of the original points to get the new points after a rotation around the origin ! 🤩
RotationOriginalNew
clockwise or counterclockwise
clockwise or counterclockwise
clockwise or counterclockwise
You can remember the formulas by imagining two arrows rotating. Try it out!
Direction:Angle of Rotation:
Try remembering the formulas for each rotation:
When you're ready, let's walk through a few examples to help us remember the rotation formulas!
Let's start by rotating the following image clockwise around the origin:

Step 1. Identify the center of rotation.

Our center of rotation is the origin, so it is the point .

Step 2. Identify the original points.

From the diagram, we can see that our original points are: .

Step 3. Subtract the center of rotation from each original point.

We're basically translating the shape to the origin so that we can use our rotation formulas.

Step 3. Identify the angle and direction of the rotation.

From the question, we know we are rotating the shape clockwise.

Step 4. Identify the formula that matches the rotation.

To do this, we start by imagining two arrows rotating clockwise:
When we rotate the arrows clockwise:
  • The axis lines up with the arrow pointing in the positive direction, so the new value is the the old value.
  • The axis lines up with the arrow pointing in the negative direction, so the new value is the negative of the old value.

Step 5. Apply the formula to each original point to get the new points.

Original PointNew Point

Step 6. Plot the new points and connect the dots.

Our new, rotated shape is the blue shape.
Nice work! Now let's try rotating this image counterclockwise around the origin:

Step 1. Identify the center of rotation.

Our center of rotation is the origin, so it is the point .

Step 2. Identify the original points.

From the diagram, we can see that our original points are: .

Step 3. Subtract the center of rotation from each original point.

We're basically translating the shape to the origin so that we can use our rotation formulas.

Step 3. Identify the angle and direction of the rotation.

From the question, we know we are rotating the shape counterclockwise.

Step 4. Identify the formula that matches the rotation.

To do this, we start by imagining two arrows rotating counterclockwise:
When we rotate the arrows counterclockwise:
  • The axis lines up with the arrow pointing in the negative direction, so the new value is the negative of the old value.
  • The axis lines up with the arrow pointing in the positive direction, so the new value is the old value.

Step 5. Apply the formula to each original point to get the new points.

Original PointNew Point

Step 6. Plot the new points and connect the dots.

Our new, rotated shape is the blue shape.
Amazing! Now let's try one more. Rotate this shape clockwise around the origin:

Step 1. Identify the center of rotation.

Our center of rotation is the origin, so it is the point .

Step 2. Identify the original points.

From the diagram, we can see that our original points are: .

Step 3. Subtract the center of rotation from each original point.

We're basically translating the shape to the origin so that we can use our rotation formulas.

Step 3. Identify the angle and direction of the rotation.

From the question, we know we are rotating the shape clockwise.

Step 4. Identify the formula that matches the rotation.

To do this, we start by imagining two arrows rotating clockwise:
When we rotate the arrows clockwise:
  • The axis lines up with the arrow pointing in the negative direction, so the new value is the negative of the old value.
  • The axis lines up with the arrow pointing in the positive direction, so the new value is the old value.

Step 5. Apply the formula to each original point to get the new points.

Original PointNew Point

Step 6. Plot the new points and connect the dots.

Our new, rotated shape is the blue shape.
Nice!! Practice with some additional problems or continue to the next section for examples of how to do rotations that are NOT around the origin.
Great work! Next up: rotations NOT around the origin.

Rotations NOT around the Origin

Rotations that are not around the origin are a little trickier, but still 100% doable if we take it one step at a time.Let’s walk through some examples to help us remember the rotation formulas!
Let's start by rotating the following image clockwise around the point :

Step 1. Identify the center of rotation.

Our center of rotation is the point .

Step 2. Identify the original points.

From the diagram, we can see that our original points are: .

Step 3. Subtract the center of rotation from each original point.

We're basically translating the shape to the origin so that we can use our rotation formulas.
Original PointCenter of RotationDifference

Step 4. Identify the angle and direction of the rotation.

From the question, we know we are rotating the shape clockwise.

Step 5. Identify the formula that matches the rotation.

To do this, we start by imagining two arrows rotating clockwise:
When we rotate the arrows clockwise:
  • The axis lines up with the arrow pointing in the positive direction, so the new value is the the old value.
  • The axis lines up with the arrow pointing in the negative direction, so the new value is the negative of the old value.

Step 6. Apply the formula to each difference.

DifferenceRotated Difference

Step 7. Add the rotated difference to the center of rotation to get the new points.

Now that we've completed the rotation, we can translate the shape back to be around the center of rotation.
Rotated DifferenceCenter of RotationNew Point

Step 8. Plot the new points and connect the dots.

Our new, rotated shape is the blue shape. (For reference, the original shape is the orange shape.)
Nice job! Now, let's try rotating the following image counterclockwise around the point :

Step 1. Identify the center of rotation.

Our center of rotation is the point .

Step 2. Identify the original points.

From the diagram, we can see that our original points are: .

Step 3. Subtract the center of rotation from each original point.

We're basically translating the shape to the origin so that we can use our rotation formulas.
Original PointCenter of RotationDifference

Step 4. Identify the angle and direction of the rotation.

From the question, we know we are rotating the shape counterclockwise.

Step 5. Identify the formula that matches the rotation.

To do this, we start by imagining two arrows rotating counterclockwise:
When we rotate the arrows counterclockwise:
  • The axis lines up with the arrow pointing in the negative direction, so the new value is the negative of the old value.
  • The axis lines up with the arrow pointing in the negative direction, so the new value is the negative of the old value.

Step 6. Apply the formula to each difference.

DifferenceRotated Difference

Step 7. Add the rotated difference to the center of rotation to get the new points.

Now that we've completed the rotation, we can translate the shape back to be around the center of rotation.
Rotated DifferenceCenter of RotationNew Point

Step 8. Plot the new points and connect the dots.

Our new, rotated shape is the blue shape. (For reference, the original shape is the orange shape.)
Amazing! Last problem - let's rotate the following image clockwise around the point :

Step 1. Identify the center of rotation.

Our center of rotation is the point .

Step 2. Identify the original points.

From the diagram, we can see that our original points are: .

Step 3. Subtract the center of rotation from each original point.

We're basically translating the shape to the origin so that we can use our rotation formulas.
Original PointCenter of RotationDifference

Step 4. Identify the angle and direction of the rotation.

From the question, we know we are rotating the shape clockwise.

Step 5. Identify the formula that matches the rotation.

To do this, we start by imagining two arrows rotating clockwise:
When we rotate the arrows clockwise:
  • The axis lines up with the arrow pointing in the negative direction, so the new value is the negative of the old value.
  • The axis lines up with the arrow pointing in the positive direction, so the new value is the old value.

Step 6. Apply the formula to each difference.

DifferenceRotated Difference

Step 7. Add the rotated difference to the center of rotation to get the new points.

Now that we've completed the rotation, we can translate the shape back to be around the center of rotation.
Rotated DifferenceCenter of RotationNew Point

Step 8. Plot the new points and connect the dots.

Our new, rotated shape is the blue shape. (For reference, the original shape is the orange shape.)
If you’d like additional practice with rotations that are not around the origin, take a moment to complete the quick practice below before closing out this lesson! ⚡
When you feel like you've mastered this lesson, click for a celebration ⬇️!
Nice work, look at you go! Thanks for checking out this lesson ☺️🙏. Where to next?