Volumes by Cross Section

Learning Resource Type

Classroom Resource

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

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.

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.

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.

    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

    Resource Provider

    Other

    License Type

    CUSTOM

    Resource Provider other

    CK-12
    https://www.ck12.org/c/calculus/volume-by-cross-section/lesson/Volumes-by-Cross…

    Custom License Type (URL)

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

    Approved Date

    Owner (Author)

    habradley
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