Cross Sections of Solids
Describe the intersection of a plane with three-dimensional figures.
Students describe intersections of planes and solids, emphasizing circular and rectangular sections of cylinders and distinguishing a true slice from a tangent intersection.
Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.
Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.
Learn the concept
A plane makes a flat intersection
A cross section is the two-dimensional region formed when a plane passes through a solid. Its shape depends on the plane's direction and location, not just on the name of the solid.
Cylinder sections
For a right circular cylinder, a plane parallel to the bases produces a circle. A plane parallel to the axis and passing through the interior produces a rectangle. An oblique plane can produce an ellipse when it cuts the curved surface without reaching either base.
Possible, impossible, and tangent
A plane tangent to a cylinder's curved side meets the finite solid along a straight segment, sometimes called a line intersection in a diagram. A triangle cannot be a plane section of a right circular cylinder. Describe where every part of a proposed boundary would come from.
Worked example
A tall cylindrical candle stands upright. Describe a plane that produces a circular section, a rectangular section, and an intersection with no area.
- Place the first plane parallel to the candle's circular base and between its ends.
- Place the second plane parallel to the axis and through the candle's interior.
- Move a plane parallel to the axis outward until it only touches the curved side along one straight segment.
Parallel to a base: circle; parallel to the axis through the interior: rectangle; tangent to the curved side: a segment with no area.
Watch the lesson
Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.
Geometry: Cross Sections
UT IPSI · Kid2Kid / Texas Education AgencyConnect the orientation of a cutting plane to the two-dimensional section of a solid.
Watch on YouTubeTry it yourself
TEKSForge practice · unscored
1. A plane cuts a sphere through its center. What is the cross section?
Show answer and reasoning
A circle
The points of the sphere in that plane form a circular section. Passing through the center gives the largest such section.
2. A student claims that a plane parallel to a cylinder's base produces a rectangle. Correct the claim.
Show answer and reasoning
It produces a circle.
The circular bases establish the shape of every interior section parallel to them. A rectangular section instead comes from an interior cut parallel to the axis.
3. Can a plane passing through the interior of a right circular cylinder produce a triangle?
Show answer and reasoning
No
The curved side does not supply three straight boundary segments, and the two parallel base planes cannot supply the three sides of a triangle.
Released STAAR practice
Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.
Archived released STAAR Geometry EOC · 2013 · Item 22 · 1 point
A plane intersects a cylinder. Which of the following cannot be formed by this intersection?
- Triangle
- Line
- Rectangle
- Circle
Show STAAR answer and explanation
Triangle
A cylinder has circular bases and a curved lateral surface; a plane cannot cut out a triangular section. Parallel-to-base cuts give circles, axial cuts give rectangles, and a tangent plane can meet the lateral surface along a line segment.
Line: A tangent plane parallel to the axis meets the lateral surface along a straight generator, represented here as a line.
Rectangle: A plane through the axis of a right circular cylinder produces a rectangle.
Circle: A plane parallel to the bases produces a circle.
Learning objectives
By the end of the lesson, students can:
- Describe a cross section using the position of a cutting plane.
- Identify possible intersections of a plane with a cylinder.
- Explain why a proposed intersection is possible or impossible.
Instructional sequence
- 1
Engage
5 minShow a cylinder and ask whether every slice would look like its circular base. Collect predictions before cutting.
- 2
Investigation
12 minCompare cuts parallel to the bases, perpendicular to the bases, and oblique to the axis. Sketch the flat region exposed by each cut.
- 3
Direct instruction
10 minDistinguish a plane passing through the interior from a tangent plane touching the curved surface along a generator segment.
- 4
Guided practice
12 minSort possible and impossible plane-cylinder intersections. Require students to describe a cutting plane rather than rely on the solid's outline.
- 5
Closure
6 minComplete the exit ticket and justify an impossible polygonal section.
Misconceptions to address
Teacher move: A circular section requires a plane parallel to the circular bases. A plane parallel to the axis and through the interior produces a rectangle.
Teacher move: A view projects the solid onto a plane. A cross section contains only the points where the cutting plane and the solid meet.
Teacher move: A tangent plane can touch the curved side along a straight segment without passing through the interior. This is a degenerate intersection, not a filled cross-sectional region.