Rotation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Rotation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A rigid transformation that turns every point of a figure by a fixed angle around a fixed center of rotation.

Like a Ferris wheel turning around its center hub—every seat traces a circle, staying the same distance from the axle while sweeping through the same angle.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A rotation moves every point around the same center by the same angle.

Common stuck point: The procedure for rotation is the easy part; the trap is rotating around the wrong center. Asking "What is the center, angle, and direction of the turn?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: What is the center, angle, and direction of the turn?

Worked Examples

Example 1

easy

Rotate the point

A(3, 2)

90°

counterclockwise about the origin. Where does A end up?

Rotate A(3, 2) by 90° CCW about the origin

Answer

A' = (-2, 3)

First step

Step 1: The rule for 90° CCW rotation is

(x, y) \to (-y, x)

Full solution

2
Step 2: Apply to $A(3, 2)$ : $A' = (-2, 3)$ .
3
Step 3: Verify distance from origin is unchanged: $\sqrt{3^2+2^2} = \sqrt{13}$ and $\sqrt{(-2)^2+3^2} = \sqrt{13}$ . ✓

The 90° CCW rotation rule

(x,y)\to(-y,x)

is derived from the rotation matrix with

\theta=90°

\cos90°=0

\sin90°=1

, giving

(-y, x)

. Rotation preserves distance from the center of rotation.

Example 2

medium

Rotate the point

B(4, 0)

60°

counterclockwise about the origin. Give exact coordinates.

Example 3

easy

Rotate the segment from

A(1, 2)

B(4, 2)

180°

about the origin. Find

A'B'

Segment AB rotated 180° about the origin

Example 4

medium

Rotate the point

P(5, 2)

90°

CCW about the center

C(1, 1)

Rotate P(5, 2) by 90° CCW about C(1, 1)

Example 5

medium

Rotate the point

(2, 0)

30°

counterclockwise about the origin. Give exact coordinates.

Example 6

medium

Triangle

A(2, 1)

B(5, 1)

C(5, 3)

is rotated

90°

CCW about the origin. Find the image vertices.

Triangle ABC rotated 90° CCW about the origin

Example 7

medium

Rotate the segment from

A(0, 0)

B(3, 4)

90°

CCW about

A

. Find

B'

Rotate segment AB by 90° CCW about A(0, 0)

Example 8

hard

Find the center of rotation for the rotation that sends

A(1, 0)

A'(0, 1)

and

B(2, 0)

B'(0, 2)

Find the center of rotation that sends A→A' and B→B'

Example 9

hard

Rotate the point

(1, 0)

45°

counterclockwise about the origin. Give exact coordinates.

Example 10

hard

The point

(6, 2)

is rotated

90°

CW about an unknown center and lands at

(2, -2)

. Find the center.

P(6, 2) rotated 90° CW lands at P'(2, −2) — find the center C

Example 11

challenge

Prove that the composition of two rotations by

\alpha

and

\beta

about (possibly different) centers

C_1

and

C_2

is either a rotation by

\alpha + \beta

or a translation. State precisely when each case occurs.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

What is the image of point

(0, 5)

after a

180°

rotation about the origin?

Rotate (0, 5) by 180° about the origin

Example 2

hard

Triangle with vertices

A(1,0)

B(3,0)

C(2,2)

is rotated

90°

CCW about the point

(1,0)

. Find the new vertices.

Triangle rotated 90° CCW about A(1, 0)

Example 3

easy

Rotate

(1, 0)

90^\circ

counterclockwise about the origin.

Rotate (1, 0) by 90° CCW about the origin

Example 4

easy

Rotate

(2, 3)

180^\circ

about the origin.

Rotate (2, 3) by 180° about the origin

Example 5

easy

Does a rotation change a figure's size?

Example 6

easy

What is the center of rotation for the rule

(x,y) \to (-y, x)

Example 7

easy

Rotate

(0, 5)

90^\circ

clockwise about the origin.

Rotate (0, 5) by 90° clockwise about the origin

Example 8

easy

After rotating a figure

360^\circ

, where does it end up?

Example 9

easy

Does the center of rotation move when you rotate a figure?

Example 10

easy

90^\circ

clockwise rotation is the same as how many degrees counterclockwise?

Example 11

medium

Rotate the point

(4, 2)

270^\circ

counterclockwise about the origin.

Rotate (4, 2) by 270° CCW about the origin

Example 12

medium

Why does rotating about a different center give a different image, even for the same angle?

Example 13

medium

Triangle vertices

(1,0)

(2,0)

(1,3)

are rotated

180^\circ

about the origin. Find the images.

Triangle rotated 180° about the origin

Example 14

medium

A figure has rotational symmetry of order 4. What is the smallest rotation that maps it onto itself?

Example 15

medium

A rotation of

90^\circ

is applied twice. What single rotation is equivalent?

Example 16

medium

Does a rotation preserve orientation (clockwise/counterclockwise order of vertices)?

Example 17

medium

A point is

5

units from the center of rotation. After any rotation, how far is its image from the center?

Example 18

medium

What rotation undoes a

90^\circ

counterclockwise rotation?

Example 19

challenge

Rotate the point

(5, 2)

90^\circ

counterclockwise about the center

(1, 1)

(not the origin).

Rotate P(5, 2) by 90° CCW about center C(1, 1)

Example 20

challenge

Two reflections across lines through the origin at

30^\circ

and

50^\circ

are applied. Show the result is a rotation and find its angle.

Example 21

challenge

A regular hexagon is rotated about its center. List all rotation angles (between

0^\circ

and

360^\circ

) that map it onto itself.

Example 22

challenge

Explain why a rotation (other than

0^\circ

360^\circ

) has exactly one fixed point.

Example 23

easy

Rotate the point

(4, 1)

90°

counterclockwise about the origin.

Rotate (4, 1) by 90° CCW about the origin

Example 24

easy

Rotate the point

(-3, 5)

180°

about the origin.

Rotate (-3, 5) by 180° about the origin

Example 25

easy

Rotate

(6, 0)

270°

counterclockwise about the origin.

Rotate (6, 0) by 270° CCW about the origin

Example 26

easy

Rotate the point

(2, -7)

90°

clockwise about the origin.

Example 27

medium

Rotate the point

(3, 5)

180°

about the center

(2, 1)

Rotate (3, 5) by 180° about center (2, 1)

Example 28

medium

A figure is rotated

90°

CCW, then

90°

CCW again about the same center. What single rotation has the same effect?

Example 29

medium

Point

(4, 6)

is rotated about the origin and lands at

(-6, 4)

. What was the rotation angle?

What rotation maps P(4, 6) to P'(−6, 4)?

Example 30

medium

A regular hexagon has a center of rotational symmetry. What is the smallest angle of rotation (in degrees) that maps the hexagon to itself?

Example 31

medium

After rotating the point

(7, -2)

180°

about the center

(3, 4)

, where does it land?

Rotate P(7, −2) by 180° about center C(3, 4)

Example 32

hard

The point

(5, 3)

is rotated

90°

CCW about an unknown center

(a, b)

and lands at

(3, 7)

. Find

(a, b)

Example 33

hard

Triangle with vertices

A(0, 0)

B(4, 0)

C(4, 3)

is rotated

90°

CCW about

B

. Find the new coordinates of

A

and

C

Triangle ABC rotated 90° CCW about vertex B(4, 0)

Example 34

hard

What single rotation about the origin equals the composition of a

250°

CCW rotation followed by a

190°

CCW rotation about the origin?

← Back to Rotation Formula Explained Practice Mode

Related Concepts

Geometric Transformation Angles Rotational Symmetry Composition of Transformations

Background Knowledge

These ideas may be useful before you work through the harder examples.

transformation geoangles