Slice an ideal cube with a plane parallel to its bottom face. The intersection is a filled square, not the whole cube and not its shadow. Move the plane without tilting it: every interior section is congruent. Now tilt the plane. The section can change even though the cube has not changed.
A section means all points belonging both to the solid and to the cutting plane. A projection instead sends points along viewing directions onto a screen; several different depths can collapse onto one projected point. A silhouette or perspective outline is therefore not automatically a section. Start every section argument by naming the solid and the exact plane.
Predict
A vertical plane through opposite vertical edges of a cube cuts a rectangle. Predict its width relative to the cube’s side. The width is a diagonal of the bottom square, not an edge length. If the cube side is s, Pythagoras gives width s√2 and vertical height s.
Predict a different section made by clipping a corner. A plane joining three points at the same distance along the three edges from that corner cuts an equilateral triangle. To justify the word equilateral, calculate distances; a perspective picture alone cannot certify them.
Experiment
The diagram is a vertical central view of a sphere of radius 5 cm. A horizontal plane is 3 cm above its center. In this view the plane appears as a line and its circular section appears edge-on. Label the right triangle and find the radius of the actual disk section.
The red horizontal segment is the section diameter in a central side view; the radius ρ runs from the vertical axis to either endpoint. The actual planar section is a filled disk.
Pythagoras gives ρ²+3²=5², hence ρ=4 cm. Repeat at height 0 and height 5. The equatorial section has radius 5; the top section is one point, with area zero. Beyond height 5 there is no intersection.
Understand
For a sphere of radius R centered at height zero, the section at height z satisfies ρ²=R²−z². This formula requires −R≤z≤R. The nonnegative root gives a radius; the negative root is not a second physical radius. The section area is π(R²−z²), which varies with height.
Orientation matters for a cylinder too. A plane perpendicular to the axis produces a disk. A plane parallel to the axis that crosses the interior produces a rectangle. A slanted plane meeting only the curved side produces an elliptical disk. For a finite cylinder, however, the slanted plane can also meet an end cap; then the section is truncated rather than a complete ellipse. Tangent planes and cuts at an endpoint can produce degenerate cases. State these conditions instead of labeling every slanted cut an ellipse.
One section cannot identify a solid. A cylinder and a sphere can share the same middle disk while differing at other heights. Even several sampled sections leave unexamined intervals. A complete section rule, together with the positions of the sections, specifies much more than an outline. Equal section areas alone still need not specify the same shape; later Cavalieri comparisons deliberately use differently shaped sections.
Apply
In practice, describe the cutting plane, locate its intersections with edges or surfaces, then justify the resulting shape with distances, parallelism or the defining equation. For a corner cut of a coordinate cube, use nonnegative coordinates to prove that the entire claimed triangle is included and that no additional face clips it.
A labeled coordinate description is acceptable construction evidence. It must identify the plane and the section vertices or radius, not simply name a familiar shape.
Continue
Rotation reverses the viewpoint: instead of extracting a planar set from a solid, you move a planar set around an axis to generate a solid. The next lesson keeps track of every swept point, including holes and overlapping sweeps.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
A cube has side 6 cm. A plane parallel to a face passes through its interior. What is the section area in cm²?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A sphere has radius 13 cm. A plane is 5 cm from its center. What is the squared section radius in cm²?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Describe the section of a cube of side 4 by the vertical plane containing two opposite vertical edges. Give side lengths and justify the shape.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In the cube 0≤x,y,z≤2, determine the section x+y+z=1. Give its vertices and prove the shape.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A right cylinder of radius 2 and height 4 is centered at z=0, and a sphere of radius 2 has the same center. Their horizontal sections at z=0 agree. Compare their sections at z=1 and explain what the first match cannot prove.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Describe a horizontal and an oblique section of a right circular cylinder. State conditions under which the oblique section is an ellipse rather than a truncated region touching an end cap.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
