ALEX Classroom Resources

   View Standards     Standard(s): [MA2019] GEO-19 (9-12) 17 :

17. Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed.

a. Give an informal argument for the formulas for the surface area and volume of a sphere, cylinder, pyramid, and cone using dissection arguments, Cavalieri's Principle, and informal limit arguments.

b. Apply geometric concepts to find missing dimensions to solve surface area or volume problems.

[MA2019] PRE-19 (9-12) 31 :

31. Graph conic sections from second-degree equations, extending from circles and parabolas to ellipses and hyperbolas, using technology to discover patterns.

a. Graph conic sections given their standard form.

Example: The graph of ^x2/₉ + ^(y-3)2/₄=1 will be an ellipse centered at (0,3) with major axis 3 and minor axis 2, while the graph of ^x2/₉ + ^(y-3)2/₄=1 will be a hyperbola centered at (0,3) with asymptotes with slope ±3/2.

b. Identify the conic section that will be formed, given its equation in general form.

Example: 5y² - 25x²=-25 will be a hyperbola.

Remember how to compute the volume of a cylinder or prism using the cross-sectional area and length (height) of the object? If the cross-sectional area is known and constant along the height, the volume calculation is easy. But, what if the cross-sectional area changes in a known manner along the line that is the height, like it does for a cone or pyramid? How could a single method in calculus be used to determine the volume of either of these types of solids?

This informational material will explain how to calculate the volume of special solid figures, like cones, by using cross-sections from the solid figure. The three-dimensional case of Cavalieri's Principle is introduced. There are corresponding videos available. Practice questions with a PDF answer key are provided.

   View Standards     Standard(s): [MA2019] GEO-19 (9-12) 17 :

17. Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed.

a. Give an informal argument for the formulas for the surface area and volume of a sphere, cylinder, pyramid, and cone using dissection arguments, Cavalieri's Principle, and informal limit arguments.

b. Apply geometric concepts to find missing dimensions to solve surface area or volume problems.

[MA2019] PRE-19 (9-12) 31 :

31. Graph conic sections from second-degree equations, extending from circles and parabolas to ellipses and hyperbolas, using technology to discover patterns.

a. Graph conic sections given their standard form.

Example: The graph of ^x2/₉ + ^(y-3)2/₄=1 will be an ellipse centered at (0,3) with major axis 3 and minor axis 2, while the graph of ^x2/₉ + ^(y-3)2/₄=1 will be a hyperbola centered at (0,3) with asymptotes with slope ±3/2.

b. Identify the conic section that will be formed, given its equation in general form.

Example: 5y² - 25x²=-25 will be a hyperbola.

Students will test their knowledge of calculating the volume of cones using cross-sections on a graph in this interactive.

How do you find the volume of a cone given its cross-section? Consider half the cross-section of the cone where the region is formed by the lines y = 0, x = 45 and the changing standard equation.  

In this interactive, students will:

• • move the red point in order to change the standard equation of a line and the cross-section associated with it.
  • move the blue point in order to traverse through the cross-section.
  • see how the radius of the circle changes when you are traversing through the solid.