Download Article
Download Article
A rotation is a type of geometrical transformation in which the vertices of a shape are rotated at a certain angle around a fixed point (called the center of rotation). [1] X Research source In simpler terms, imagine gluing a triangle to the second hand of a clock that is spinning backwards. Usually, you will be asked to rotate a shape around the origin, which is the point (0, 0) on a coordinate plane. You can rotate shapes 90, 180, or 270 degrees around the origin using three basic formulas.
Steps
-
Note the corresponding clockwise and counterclockwise rotations. Rotating a shape 90 degrees is the same as rotating it 270 degrees clockwise. [2] X Research source The convention is that when rotating shapes on a coordinate plane, they rotate counterclockwise, or towards the left. [3] X Research source You should assume this, unless it is noted in the problem that you need to rotate clockwise.
- For example, if the problem states, “Rotate the shape 90 degrees around the origin,” you can assume you are rotating the shape counterclockwise.
- You would complete this problem the same way you complete a problem that asks “Rotate the shape 270 degrees clockwise around the origin.”
- You might also see, “Rotate this shape -270 degrees around the origin.”
- For example, if the problem states, “Rotate the shape 90 degrees around the origin,” you can assume you are rotating the shape counterclockwise.
-
Find the coordinates of the original vertices. If these aren’t already provided, determine the coordinates using the graph. Remember that coordinates of points are shown using the formula, where equals the point on the horizontal, or x-axis, and equals the point on the vertical, or y-axis. [4] X Research source
- For example, you might have a triangle with points (4, 6), (1, 2), and (1, 8).
Advertisement -
Plug the coordinates into the formula. Make sure that you keep your x and y-coordinates straight. In this formula, you take the negative of the y value, and then switch the order of the coordinates. [7] X Research source
- For example, the points (4, 6), (1, 2), and (1, 8) become (-6, 4), (-2, 1), and (-8, 1).
-
Draw the new shape. Plot the new vertex points on the plane. Connect your points using a straightedge. The resulting shape shows the original shape rotated 90 degrees around the origin. [8] X Research source
Advertisement
-
Identify the corresponding clockwise and counterclockwise rotations. Since a full rotation has 360 degrees, rotating a shape 180 degrees clockwise is the same as rotating 180 counterclockwise.
- If the problem states, “Rotate the shape 180 degrees around the origin,” you can assume you are rotating the shape counterclockwise.
- You would complete this problem the same way you complete a problem that asks “Rotate the shape 180 degrees clockwise around the origin.”
- You might also see, “Rotate this shape -180 degrees around the origin.”
- If the problem states, “Rotate the shape 180 degrees around the origin,” you can assume you are rotating the shape counterclockwise.
-
Write down the coordinates of the original shape’s vertices. These will likely be given. If not, you should be able to deduce them from looking at the coordinate graph. Remember to note the coordinates of each vertex’s point using the (x, y) convention.
- For example, you might have a rhombus with points (4, 6), (-4, 6), (-2, -1), and (2, -1).
-
Plug the coordinates into the formula. Take care to plug the correct coordinate into the correct position of the new ordered pair. In this formula, you keep the x and y values in the same position, but you take the negative value of each coordinate.
- For example, the points (4, 6), (-4, 6), (-2, -1), and (2, -1) become (-4, -6), (4, -6), (2, 1), and (-2, 1).
-
Draw the new shape. Plot the new vertex points on the plane. Connect your points using a straightedge. The resulting shape shows the original shape rotated 180 degrees around the origin. [11] X Research source
Advertisement
-
Note the corresponding clockwise and counterclockwise rotations. Rotating a shape 270 degrees is the same as rotating it 90 degrees clockwise. Conventionally, shapes are rotated counterclockwise on a coordinate plane. [12] X Research source You should assume this, unless it is noted in the problem that you need to rotate clockwise.
- For example, if the problem states, “Rotate the shape 270 degrees around the origin,” you can assume you are rotating the shape counterclockwise.
- You would complete this problem the same way you complete a problem that asks “Rotate the shape 90 degrees clockwise around the origin.”
- You might also see, “Rotate this shape -90 degrees around the origin.”
- For example, if the problem states, “Rotate the shape 270 degrees around the origin,” you can assume you are rotating the shape counterclockwise.
-
Find the coordinates of the original vertices. This information should be provided, or you should be able to easily find the coordinates by looking at the coordinate plane.
- For example, you might have a triangle with points (4, 6), (1, 2), and (1, 8).
-
Plug the coordinates into the formula. Make sure you plug the correct x and y values into the new coordinate pair. In this formula, the x and y values are reversed, and you take the negative value of the x coordinate. [15] X Research source
- For example, the points (4, 6), (1, 2), and (1, 8) become (6, -4), (2, -1), and (8, -1).
-
Draw the new shape. Draw the new points on the plane. Use a straightedge to connect them. The resulting shape shows the original shape rotated 270 degrees around the origin. [16] X Research source
Advertisement
Community Q&A
Search
-
QuestionHow would I rotate a quadrilateral about a point?NitakuarCommunity AnswerIf the point is the origin, then simply apply above the transformation for each of the vertices. If it is some other point, then shift the coordinate axes parallel to the original so it becomes the origin and, after rotation, revert back to the original coordinates.
-
QuestionWhat does "around the origin" mean? Does it mean around point (0,0)?Community AnswerThe origin is around (0,0), the center of the diagram.
-
QuestionWhat if the number is negative?DonaganTop AnswererIf we assume a positive rotation is counterclockwise (anti-clockwise), a negative rotation would be clockwise (to the right at the top).
Ask a Question
200 characters left
Include your email address to get a message when this question is answered.
Submit
Advertisement
Tips
Submit a Tip
All tip submissions are carefully reviewed before being published
Name
Please provide your name and last initial
Thanks for submitting a tip for review!
References
- ↑ http://www.varsitytutors.com/hotmath/hotmath_help/topics/rotations
- ↑ http://www.virtualnerd.com/pre-algebra/geometry/transformations-symmetry/rotating-figures/rotate-270-degrees-about-origin
- ↑ https://www.khanacademy.org/math/geometry/hs-geo-transformations/hs-geo-rotations/v/points-after-rotation
- ↑ https://www.khanacademy.org/math/geometry/hs-geo-transformations/hs-geo-rotations/a/rotating-shapes
- ↑ https://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGTransformationRotation.html
- ↑ http://www.virtualnerd.com/pre-algebra/geometry/transformations-symmetry/rotating-figures/rotate-270-degrees-about-origin
- ↑ https://www.khanacademy.org/math/geometry/hs-geo-transformations/hs-geo-rotations/a/rotating-shapes
- ↑ https://www.khanacademy.org/math/geometry/hs-geo-transformations/hs-geo-rotations/a/rotating-shapes
- ↑ https://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGTransformationRotation.html
- ↑ http://www.virtualnerd.com/pre-algebra/geometry/transformations-symmetry/rotating-figures/rotate-180-degrees-about-origin
- ↑ https://www.math-only-math.com/180-degree-rotation.html
- ↑ https://www.khanacademy.org/math/geometry/hs-geo-transformations/hs-geo-rotations/v/points-after-rotation
- ↑ https://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGTransformationRotation.html
- ↑ https://virtualnerd.com/pre-algebra/geometry/transformations-symmetry/rotating-figures/rotate-270-degrees-about-origin
- ↑ https://virtualnerd.com/pre-algebra/geometry/transformations-symmetry/rotating-figures/rotate-270-degrees-about-origin
- ↑ https://virtualnerd.com/pre-algebra/geometry/transformations-symmetry/rotating-figures/rotate-270-degrees-about-origin
About This Article
Article Summary
X
To rotate a shape 90 degrees around the point of origin, turn the x and y coordinates into -y and +x coordinates. For example, a triangle with the coordinates 1,2, 4,2, and 4,4 would become -2,1, -2,4, and -4,4. If you want to rotate a shape 180 degrees around the point of origin, turn the x and y coordinates into -y and -x coordinates. So, if a line has the coordinates 2,4 and 4,5, it would rotate to -4,-2 and -5,-4. Read more to learn how to rotate a shape 270 degrees!
Did this summary help you?
Thanks to all authors for creating a page that has been read 169,493 times.
Reader Success Stories
- "I have an upcoming Maths test, and one of the chapters, which is going to come out, is rotating shapes. Even though the teacher repeated the steps many times to me, I struggled to remember. Now I understand it. Wish me well for the test!" ..." more
Advertisement
"I have an upcoming Maths test, and one of the chapters, which is going to come out, is rotating shapes. Even though the teacher repeated the steps many times to me, I struggled to remember. Now I understand it. Wish me well for the test!"
..."
more
"This really helped me with my math project. I was having trouble, so I looked it up and found this website. So helpful, I understood so well!"
..."
more
"Was having trouble on some homework, looked it up, found this website, and used it. Great information!"
"This really helped because it provided a useful way to rotate shapes without using another tool. "
"While self learning, I was having trouble with rotations, but this really helped. Thank you!"
