Rotation of Two-Dimensional Solids Module 14 Lesson 2 In this lesson, you will learn to identify the three-dimensional solid generated from the rotation of a two-dimensional shape. In simpler terms, what 2D shape do we use to create a 3D shape?

Review of Two-Dimensional and Three-Dimensional Figures A Two-Dimensional (2D) shape is a shape that only has two dimensions: width and height. Examples: Squares, Circles, Triangles, etc are two dimensional objects A Three-Dimensional (3D) shape is a shape that has three dimensions: width, depth and height. Examples: Cube, Cylinder, etc are three dimensional objects Lets do a quick review of 2D and 3D figures.

"Rotation" means turning around a center. 2D 3D Three-Dimensional figures can be generated by rotating two-dimensional figures. "Rotation" means turning around a center. A three-dimensional object rotates always around an imaginary line called a rotation axis. Two-dimensional figures can be transformed through space to create three-dimensional solids. One way is to rotate a two-dimensional figure is about a line. This line is sometimes called an axis of symmetry. The key feature of any rotation is that they are symmetrical about the line or axis you are rotating them about. Imagine if begin to twirl my pencil around and around and around. What 3d shape have I produced?

WHAT DID YOU GUESS? If you guessed cone, you are correct! A cone is solid revolution of a right triangle around one of its legs.

WHAT 3D SHAPE IS PRODUCED IF WE ROTATE A SEMI-CIRCLE ? A sphere is solid revolution of a semi-circle around its diameter.

You try: Given the shape below, determine the 3D solid formed by rotating the two-dimensional shape about the line given. 1. 2. If you are having trouble visualizing it, grab a pencil, cut the shape out from a piece of paper and tape it to the pencil and spin!

You try: Given the shape below, determine the 3D solid formed by rotating the two-dimensional shape about the line given. 1. 2. If you are having trouble visualizing it, grab a pencil, cut the shape out from a piece of paper and tape it to the pencil and spin! A square rotated about the above line results in a right cylinder. A circle rotated about the above line results in a sphere.

Another Way to Visualize the Rotation If you do not want to cut out the shapes, and you still need help visualizing the rotation: Draw your shape and shade the region to be rotated. Next, draw a reflection (mirror image) of the region about the axis or line of rotation. Connect the vertices of the original image and its reflection using curved lines If we don’t want to use a manipulative such as a pencil to visualize these revolutions, we can also draw a graph. Step 1: Step 2: Step 3:

Word Problems Volume = 3.14r2 h Volume = 3.14(5)2 (10) Volume ≈ 785 A square with area of 100 cm2 is rotated to form a cylinder. What is the volume of the cylinder? Area = 100 cm2 s2 = 100 s = 10 10 10 Last but not least, let’s look at a few word problems. A cylinder is created when you revolve a square around on its side. Each side is 10. The area of a cylinder is 3.14r2 h = 3.14(5)2 10 = 785 Volume = 3.14r2 h Volume = 3.14(5)2 (10) Volume ≈ 785

Word Problems Given a cone with a radius of 6 ft and a height of 12 ft, find the area of the triangle formed by a perpendicular cross-section down through the cone's center. 12 Problem # 2 A cone can be generated by twirling a right triangle around one of its legs.  A= ½ bh = ½ (6)(12) = 36 Now it’s time for you to try! Please proceed to the Check the Your Knowledge Activity. There is also a PDF version of this video for you to print. 6 A= ½ bh A = ½ (6)(12) A = 36 square ft