Classroom Resource

Volume by Cross Section: Volume of the Cone Interactive

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

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.
Mathematics (2019) Grade(s): 09-12 - Geometry with Data Analysis

MA19.GDA.17

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

Unpacked Content

UP:MA19.GDA.17

Vocabulary

  • Dissection arguments
Principle
  • Cylinder
  • Pyramid
  • Cone
  • Ratio
  • Circumference
  • Parallelogram
  • Limits
  • Conjecture
  • Cross-section
  • Surface Area

    • Knowledge

    Students know:
    • Techniques to find the area and perimeter of parallelograms,Techniques to find the area of circles or polygons

    Skills

    Students are able to:
    • Accurately decompose circles, spheres, cylinders, pyramids, and cones into other geometric shapes.
    • Explain and justify how the formulas for surface area, and volume of a sphere, cylinder, pyramid, and cone may be created from the use of other geometric shapes.

    Understanding

    Students understand that:
    • Geometric shapes may be decomposed into other shapes which may be useful in creating formulas.
    • Geometric shapes may be divided into an infinite number of smaller geometric shapes, and the combination of those shapes maintain the area and volume of the original shape.
    Mathematics (2019) Grade(s): 09-12 - Precalculus

    MA19.PRE.31

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

    Unpacked Content

    UP:MA19.PRE.31

    Vocabulary

    • Hyperbola
    • Ellipse
    • Degenerate conic
    • Focus (foci)
    • Latus rectum (focal distance)
    • Major axis (transverse axis)
    • Minor axis (conjugate axis)
    • Eccentricity
    • Asymptote
    • Directrix
    • Locus

    Knowledge

    Students know:
    • Vertex form of a parabola.
    • Standard form of a circle.
    • Vertex and axis of symmetry of a parabola.
    • Completing the square.
    • Factoring a quadratic function.

    Skills

    Students are able to:
    • Graph equations of parabolas.
    • Graph equations of circles.
    • Graph equations of ellipses.
    • Calculate eccentricities of ellipses.
    • Graph equations of hyperbolas.
    • Classify a conic section using its general equation and/or its discriminant.

    Understanding

    Students understand that:
    • A conic section is a graph of an equation of the form Ax2 + Bxy + Cy2 +Dx +Ey + F = 0.
    • The only conic sections that are functions are parabolas that open upward or downward, previously learned as quadratic functions and hyperbolas that are written in the form of a rational function.
    • Using algebra to manipulate the equation of a conic section, particularly the method of "completing the square"" can be used to determine the parts and properties of its graph
    Link to Resource
    https://www.ck12.org/assessment/tools/geometry-tool/plix.html?eId=MAT.CAL.304.2…

    Resource Provider

    Other

    Resource Provider other

    CK-12
    Accessibility
    License

    License Type

    CUSTOM

    Custom License Type (URL)

    https://www.ck12info.org/terms-of-use/

    Comments or Notes

    Students must create a free account online to access the interactive.

    Approved Date

    habradley
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