Secondary Math Two and Three-Dimensional Objects

Warm Up – Evaluate each sum (Look up formulas in 3.1 and 3.2

Slicing or cutting through a three-dimensional figure with a plane can create a two-dimensional shape. For instance, slicing through a cone can create a triangle, circle, parabola, or ellipse. A cone is a three- dimensional figure that has a circle base and a vertex that is not in the same plane as the base. The height of the cone is the perpendicular distance between the vertex and the base.

Slicing or cutting through a three- dimensional figure.  Slicing a cone through the vertex creates a triangle.

Slicing or cutting through a three- dimensional figure.  Slicing a cone parallel to the base creates a circle.

Slicing or cutting through a three- dimensional figure.  Slicing a cone diagonally, through the base, creates a parabola.

Slicing or cutting through a three- dimensional figure.  Slicing a cone diagonally creates an ellipse.

Videos     

Facts  Whenever a slice is made parallel to the base of the three- dimensional object then the two-dimensional cross-section created will be similar to the base.  The maximum number of sides that a two-dimensional cross-section can have is equal to the number of faces of the three-dimensional figure from which it is sliced.  The two-dimensional cross section will have the same number of sides as the number of intersected faces of the solid.

Rotating a two-dimensional figure around an axis creates a three dimensional figure.  Start with a rectangle that has a side on each axis.  What is the area of this shape?

Rotating a two-dimensional figure around an axis creates a three dimensional figure.  Rotating around the y-axis creates a right circular cylinder with a height of y and radius of x.

Rotating a two-dimensional figure around an axis creates a three dimensional figure.  What is the height?  What is the radius?  What is the volume? Volume of a prism is the area of the base times the height.

Rotating a two-dimensional figure around an axis creates a three dimensional figure.  Rotating around the x- axis creates a right circular cylinder with height x and radius y.  Notice that the side perpendicular to the axis of rotation is flat, while the side parallel is curved.

Rotating a two-dimensional figure around an axis creates a three dimensional figure.  What is the height?  What is the radius?  What is the volume?

Rotating a two-dimensional figure around an axis creates a three dimensional figure.  Rotating a rectangle that has only one side on an axis creates a cylinder with a hole in the middle or a doughnut.

Rotating a two-dimensional figure around an axis creates a three dimensional figure.  What is the height?  What is the volume of the figure?

Video  figures figures

Formulas: Rectangular Prism Cylinder Cone Sphere

Homework #4-7  4. Horizontal slice.  5. Vertical slice through the vertex opposite the base.  6. Vertical slice not through the vertex opposite the base.  7. Diagonal slice through all four lateral sides and the base.

Homework Part B #3  What is the area of the shape?  Describe and sketch the solid created by rotating the shape around the y-axis.  What is the volume of the solid?