identify the shapes of cross-sections of prisms, pyramids, cylinders, cones and spheres, and identify the solid generated by rotating a plane figure about a line.
Instruction
A cross-section is the flat shape you get when a plane slices through a solid. Think of cutting a loaf of bread, a carrot or a block of cheese: the exposed face is the cross-section. Doctors read CT and MRI scans as stacks of cross-sections, and engineers slice a 3D design to plan how a 3D printer will build it layer by layer.
Slicing a cube. A plane parallel to a face gives a square. A plane cutting straight down through two opposite edges gives a rectangle. A plane that cuts off a corner gives a triangle; if it passes through three vertices adjacent to one corner, the triangle is equilateral, because its sides are diagonals of congruent square faces. Most surprisingly, a plane through the center perpendicular to a long diagonal crosses six faces and gives a regular hexagon, with vertices at the midpoints of six edges.
Why a hexagon, and why regular? The plane meets six edges at their midpoints, and each side of the hexagon joins the midpoints of two edges of one square face, so each side is half a face diagonal. All six sides are equal. The symmetry of the cube about that diagonal (rotations by 120°) carries the hexagon onto itself, which is the Geometry A symmetry argument applied in space.
Other solids.
- Cylinder: parallel to the base, a circle; through the axis, a rectangle; tilted, an ellipse.
- Cone: parallel to the base, a smaller circle (a dilation of the base, centered at the apex); through the apex, a triangle; tilted, an ellipse or, parallel to a slant side, a parabola.
- Sphere: every plane section is a circle, largest through the center.
- Square pyramid: parallel to the base, a smaller square; through the apex and perpendicular to the base, a triangle.
Solids of revolution. Rotate a plane figure 360° about a line in its plane and it sweeps out a solid. A rectangle rotated about one side makes a cylinder. A right triangle rotated about a leg makes a cone. A semicircle rotated about its diameter makes a sphere. A rectangle rotated about a line a short distance away makes a cylinder with a cylindrical hole, like a washer or a pipe. A potter’s wheel, a lathe and a spinning ride all make solids of revolution.
Modeling with shapes (G-MG.1). Describing objects this way is how you start to compute with them: a tree trunk is roughly a cylinder, a traffic cone a cone, a grain silo a cylinder topped by a hemisphere. Which cross-section would you measure to estimate the volume of wood in a log?
Reasoning, not guessing. For each cross-section claim, say which edges or faces the plane crosses. The number of sides of a polygonal cross-section equals the number of faces the plane cuts through, which is why a cube (six faces) can never give a heptagon.
Formative check
Work through these before moving on. They are not graded — they tell you, and your teacher, whether the standard below has landed yet.
A plane cuts a cube through its center, perpendicular to a long diagonal. What shape is the cross-section?
Which solid is generated by rotating a right triangle 360° about one of its legs?
A plane section of a cube can be a heptagon (seven sides).
Simulation & tools
Slice a cube in GeoGebra 3D
Open a blank GeoGebra 3D Calculator and type these commands in the input bar one at a time:
c = Cube((0,0,0),(4,0,0),(4,4,0))p: x + y + z = 6(this creates a plane) and thenIntersectPath(p, c), which draws the cross-section. Rotate the view and record the shape and the number of sides.- Redefine the plane (double-click it in the Algebra view) as
z = 2, thenx + y = 4, thenx + y + z = 2, and record each cross-section. - Find a plane that gives a rectangle that is not a square, and one that gives a pentagon.
Answer: For each plane, name the shape and say which faces of the cube the plane crosses. Explain why the plane x + y + z = 6 gives six equal sides.
Without a device: use a cube of clay, foam or a cut potato and a knife (with an adult), or draw the cube in oblique projection and shade each section.
Open GeoGebra ↗ · GeoGebra · free for non-commercial use
How confident are you that you can identify cross-sections of common solids and the solids generated by rotating plane figures?
Practice
Work these on paper or in your notebook, then open Check your answer. Aim for all of Fluency and Application; try at least one Challenge.
Printable version: this unit’s practice workbook (PDF)
Fluency
Build speed and accuracy with the core skill.
- Name the cross-section of a cube cut by a plane parallel to one face.
Check your answer
Answer: A square congruent to the face. - Name the cross-section of a cylinder cut by a plane through its axis.
Check your answer
Answer: A rectangle. - Name the cross-section of a cone cut by a plane parallel to its base.
Check your answer
Answer: A circle, smaller than the base. - Name the cross-section of a square pyramid cut by a vertical plane through the apex and the midpoints of two opposite base edges.
Check your answer
Answer: An isosceles triangle. - What solid is formed by rotating a rectangle 360° about one of its sides?
Check your answer
Answer: A cylinder. - What solid is formed by rotating a semicircle 360° about its diameter?
Check your answer
Answer: A sphere. - What is the greatest number of sides a plane section of a triangular prism can have? Explain.
Check your answer
Answer: 5, because a triangular prism has 5 faces and each side of the section lies in a different face. - Every plane section of a sphere is what shape, and where is the largest one?
Check your answer
Answer: A circle; the largest passes through the center (a great circle).
Application
Use the skill in context. Show your reasoning.
- A log is modeled as a cylinder of radius 0.3 m and length 4 m. What cross-section would a sawmill measure to estimate its volume, what is its area, and what is the volume?
Check your answer
Answer: A circular cross-section of area 0.09π ≈ 0.283 m2; volume 0.36π ≈ 1.131 m3Area π(0.3)2; volume = cross-section area × length. - A 3 cm by 5 cm rectangle is rotated about its 5 cm side. Name the solid and find its volume.
Check your answer
Answer: A cylinder of radius 3 cm and height 5 cm; 45π ≈ 141.37 cm3 - A right triangle with legs 6 cm and 4 cm is rotated about the 6 cm leg. Name the solid and find its volume.
Check your answer
Answer: A cone of radius 4 cm and height 6 cm; 32π ≈ 100.53 cm3⅓π(4)2(6).
Challenge
Stretch problems. Expect to think before you write.
- A cube has edge 4. Find the exact area of (a) the triangular cross-section through the three vertices adjacent to one corner and (b) the hexagonal cross-section through the midpoints of six edges.
Check your answer
Answer: (a) 8√3 ≈ 13.86; (b) 12√3 ≈ 20.78(a) The triangle is equilateral with side 4√2 (face diagonals): area (√3/4)(32). (b) The hexagon is regular with side half a face diagonal, 2√2: area 6 × (√3/4)(8).
Review
Keep earlier skills sharp.
- In a right triangle, an acute angle is 32° and the adjacent leg is 38. Find the opposite leg. Round to the nearest tenth.
Check your answer
Answer: 23.7tan 32° = opp/38 - A 45°-45°-90° triangle has legs of length 5. Find the hypotenuse.
Check your answer
Answer: 5√2In a 45-45-90 triangle the hypotenuse is √2 times a leg. - Triangle ABC is similar to triangle DEF. AB = 12 and the corresponding side DE = 6. If BC = 20, find EF, and the ratio of the triangles’ areas.
Check your answer
Answer: EF = 10; area ratio (DEF : ABC) = 1/4The scale factor is 6/12 = 1/2; areas scale by its square.