Identify the shapes of two-dimensional cross-sections of three-dimensional figures.
Clarifications
Clarification 1: Instruction includes the use of manipulatives and models to visualize cross-sections.
Clarification 2: Instruction focuses on cross-sections of right cylinders, right prisms, right pyramids and right cones that are parallel or perpendicular to the base.
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Geometric Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Circle
- Cone
- Cylinder
- Prism
- Pyramid
- Rectangle
- Square
- Triangle
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In the elementary grades, students classified two- and three-dimensional figures. In middle
grades, students worked with nets of three-dimensional figures when determining surface area.
In Geometry, students begin to understand the concept of the two-dimensional cross-sections of
familiar three-dimensional figures in preparation to discuss the formulas for the volume of these
figures. In later courses, students will relate cross-sections to conic sections and will use cross-sections as a basis for finding the volumes of other three-dimensional figures.
- For the purposes of this benchmark, there is no expectation for students to master cross-sections of non-right, or oblique, three-dimensional figures. Instruction can include these
as an enrichment or as comparison to cross-sections of right figures. As an additional
enrichment, composite figures can also be utilized within instruction.
- Instruction includes the student understanding that among the shapes of the two-dimensional cross-sections can be circles, rectangles, squares, triangles, trapezoids and all
other polygons that can be used as the base of right prisms and right pyramids (when the
plane is parallel or perpendicular to the bases).
- When the vertical cross-section of a right cone does not include the apex, it is one piece
of a hyperbola. Since many students at this level are not familiar with hyperbolas, the
expectation is not to name the hyperbola, but be able to draw or visualize this cross-section. For enrichment, instruction may include showing that it is not a parabola, since it
has diagonal asymptotes.
- Instruction focuses on pyramids with bases that are either equilateral triangles or squares,
with the vertical cross-sections being parallel to a side of the base so that the vertical
cross-sections are isosceles triangles and isosceles trapezoids. Since vertical cross-sections of pyramids include triangles that are not isosceles, and a variety of irregular
polygons it may be difficult for students to visualize or name.
- Instruction includes utilizing objects, such as soda cans, cereal boxes or party hats, as
models to explore their cross-sections. Additionally, students can explore other cross-sections using manipulatives such as clay and string to cut through the three-dimensional
figure. (MTR.7.1)
Common Misconceptions or Errors
- Students may oversimplify when they try to visualize cross-sections. To help address this
misconception, use real-world three-dimensional figures to explore their cross-sections,
as well as animations.
- Students may have difficulty with vertical cross-sections of pyramids and cones. To help
address this, utilize manipulatives and physical models within instruction.
Instructional Tasks
Instructional Task 1 (MTR.3.1, MTR.4.1) - Part A. Draw and name right three-dimensional figures that could have a triangular cross-section.
- Part B. Draw and name right three-dimensional figures that could have a circular cross-section.
- Part C. Compare your answers from Parts A and B with a partner.
Instructional Task 2 (MTR.3.1) - Part A. Fill in the blank below.
Both a right cylinder and a right prism have _______ cross-sections when cut
perpendicular to the base.
- Part B. Draw some cross-sections that are perpendicular to the base for each figure below.
Instructional Items
Instructional Item 1 - Which of the following polygons are cross-sections that are parallel or perpendicular to the
base of a regular pentagonal pyramid? Select all that apply.
- a. Triangle
- b. Parallelogram
- c. Trapezoid
- d. Pentagon
- e. Hexagon
- f. Octagon
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
Related Access Points
Access Point Number |
Access Point Title |
MA.912.GR.4.AP.1 | Identify the shape of a two-dimensional cross section of a three-dimensional figure. |
Related Resources
Formative Assessments
Name |
Description |
Inside the Box | Students are asked to identify and draw cross sections of a rectangular prism and to describe their dimensions. |
Slice of a Cone | Students are asked to sketch, describe, and compare three horizontal cross sections of a cone. |
Slice It | Students are asked to identify and describe two-dimensional cross sections of three-dimensional solids. |
Mudslide | Students are asked to create a model to estimate volume and mass. |
Square Pyramid Slices | Students are asked to sketch and describe the two-dimensional figures that result from slicing a square pyramid. |
Rectangular Prism Slices | Students are asked to sketch and describe two-dimensional figures that result from slicing a rectangular prism. |
Cylinder Slices | Students are asked to sketch and describe the two-dimensional figures that result from slicing a cylinder. |
Cone Slices | Students are asked to sketch and describe the two-dimensional figures that result from slicing a cone. |
Working Backwards – 2D Rotations | Students are given a solid and asked to determine the two-dimensional shape that will create the solid when rotated about the y-axis. |
Lesson Plans
Name |
Description |
Exploring Cavalieri's Principle | Students will explore Cavalieri's Principle using technology. Students will calculate the volume of oblique solids and determine if Cavalieri's Principle applies.
Students will also perform transformations of a base figure in a 3-dimensional coordinate system to observe the creation of right and oblique solid figures. After these observations, students will create a conjecture about calculating the volume of the oblique solids. Students will use the conjecture to determine situations in which Cavalieri's Principle applies and then calculate the volume of various oblique solids. |
Plane Slice | Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Students will use modeling clay to explore the cross sections that result from slicing a 3-dimensional figure. |
Original Student Tutorial
Name |
Description |
Ninja Nancy Slices | Learn how to determine the shape of a cross-section created by the intersection of a slicing plane with a pyramid or prism in this ninja-themed, interactive tutorial. |
Perspectives Video: Professional/Enthusiasts
Problem-Solving Tasks
Name |
Description |
Global Positioning System II | Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems. |
Tennis Balls in a Can | This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder |
Student Resources
Original Student Tutorial
Name |
Description |
Ninja Nancy Slices: | Learn how to determine the shape of a cross-section created by the intersection of a slicing plane with a pyramid or prism in this ninja-themed, interactive tutorial. |
Problem-Solving Tasks
Name |
Description |
Global Positioning System II: | Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems. |
Tennis Balls in a Can: | This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder |
Parent Resources
Problem-Solving Tasks
Name |
Description |
Global Positioning System II: | Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems. |
Tennis Balls in a Can: | This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder |