Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify the three-dimensional objects generated by rotating two-dimensional figures

Learning goal: Identify the shapes of two-dimensional cross-sections of three-dimensional objects such as cubes, prisms, cylinders, cones, spheres and pyramids, depending on how the cutting plane is placed, and identify the three-dimensional objects generated by rotations of two-dimensional objects about a line.

Cross-sections: what they are. A cross-section of a solid is the two-dimensional figure where a plane cuts through it, the flat face you would see if the solid were sliced along that plane. The shape depends on the solid and on the position and tilt of the plane, so the honest answer to "what is the cross-section of a cylinder?" is "it depends on the cut". Three families of cuts organize most cases: planes parallel to the base, planes perpendicular to the base (for solids with an axis, the ones through the axis are the most important), and oblique planes at other angles.

Cube and rectangular prism. A plane parallel to a face of a cube of side 6 cm cuts a square of side 6 cm, at any height. A plane perpendicular to the base face (a vertical plane) that crosses the cube cuts a rectangle of height 6 cm whose width is the chord the plane makes across the base square: 6 cm when the plane is parallel to a side face, and up to 6√2 ≈ 8.49 cm when it runs along a diagonal of the base. A plane that slices off one corner, cutting three edges that meet at a vertex, gives a triangle. Planes can also give pentagons and hexagons: the plane through the midpoints of six edges, perpendicular to a space diagonal at the center, cuts a regular hexagon whose side is half a face diagonal, 3√2 ≈ 4.24 cm. What a plane can never cut from a cube is a circle or any curved figure, because the intersection of a plane with a flat face is a straight segment, so every cross-section of a convex polyhedron such as a cube is a polygon with at most one side per face: a cube's cross-sections have at most 6 sides.

Cylinder. For a right circular cylinder of radius 4 cm and height 10 cm, a plane parallel to the bases cuts a circle of radius 4 cm, area 16π cm². A plane containing the axis cuts a rectangle 8 cm by 10 cm, area 80 cm². A plane parallel to the axis but 2 cm from it cuts a narrower rectangle: its width is the chord of the base circle at distance 2 cm from the center, 2√(16 − 4) = 2√12 ≈ 6.93 cm, so the rectangle is about 6.93 cm by 10 cm, area 40√3 ≈ 69.28 cm². A tilted plane that passes through the curved surface only, not the bases, cuts an ellipse. A tilted cut that also crosses one base gives a piece of an ellipse closed by one straight edge; a steeper cut that crosses both bases has two straight edges joined by two curved ones.

Cone. A plane parallel to the base of a right circular cone cuts a circle, smaller the closer it is to the apex; for a cone of radius 5 cm and height 12 cm the cut 3 cm below the apex has radius 5 × 3/12 = 1.25 cm. A plane through the apex and the axis cuts an isosceles triangle with base 10 cm and height 12 cm. A tilted plane that misses the base and the apex cuts an ellipse; a plane parallel to a slant edge cuts along a parabola; a plane parallel to the axis but not through it cuts along one branch of a hyperbola. In a solid cone with a base, the parabola and hyperbola cuts are pieces of those curves closed off by a straight segment across the base. These are the conic sections, and they come from cutting the same cone at different tilts.

Sphere. Every plane that meets a sphere in more than a point cuts a circle. The circle is largest, a great circle of radius r, when the plane passes through the center. A plane at distance d from the center of a sphere of radius r cuts a circle of radius √(r² − d²): for r = 13 cm and d = 5 cm the radius is √(169 − 25) = 12 cm and the area is 144π cm². This is the same right triangle used in the sphere volume argument.

Pyramid. A plane parallel to the base of a right square pyramid (apex above the center of the base) cuts a smaller square. A pyramid with base 10 cm and height 12 cm cut 3 cm above the base, 9 cm below the apex, gives a square scaled by 9/12 = 3/4, side 7.5 cm and area 56.25 cm². A plane through the apex perpendicular to the base cuts a triangle. A plane perpendicular to the base and parallel to one base edge that misses the apex cuts a trapezoid, with two parallel sides where it crosses the base and the lateral face on that edge's side. Other vertical planes that miss the apex give a triangle (near a corner) or a quadrilateral that is not a trapezoid, so the direction of the cut matters. A vertical plane never cuts a pentagon from this pyramid. Above its segment across the base, the top edge of the section bends only where the plane crosses a slanted edge of the pyramid. Seen from above, those slanted edges lie along the two diagonals of the base, and a line crosses each diagonal at most once. So the top edge has at most three straight pieces, and with the base segment the section has at most four sides. A tilted (oblique) plane that crosses the base and all four lateral faces does cut a pentagon.

Solids of rotation. Rotate a two-dimensional figure a full turn about a line in its plane, the axis, and the figure sweeps out a solid of rotation. Every point of the figure that is off the axis traces a circle centered on the axis, while points on the axis stay where they are. So each cross-section perpendicular to the axis is made of circles about the axis: a disc where the figure reaches the axis at that position, and a ring (an annulus) where it does not, as in the tube below. Identifying the solid means asking what those discs and rings look like along the axis. A rectangle rotated about one of its sides generates a cylinder: 3 cm by 5 cm rotated about the 5 cm side gives radius 3 cm and height 5 cm, volume 45π ≈ 141.37 cm³, while rotation about the 3 cm side gives radius 5 cm and height 3 cm, volume 75π ≈ 235.62 cm³, a different solid from the same rectangle. A rectangle rotated about a line parallel to a side but not touching it generates a tube, a cylinder with a cylindrical hole. A right triangle rotated about a leg generates a cone with that leg as height and the other leg as radius: legs 6 cm and 8 cm about the 8 cm leg give radius 6, height 8, volume 96π ≈ 301.59 cm³; about the 6 cm leg, radius 8, height 6, volume 128π ≈ 402.12 cm³. A right triangle rotated about its hypotenuse generates two cones joined base to base. A semicircle rotated about its diameter generates a sphere; radius 4 cm gives (4/3)π × 64 = 256π/3 ≈ 268.08 cm³. A circle rotated about a line outside it generates a torus, a ring shape. A right trapezoid rotated about its perpendicular leg generates a frustum. An isosceles triangle rotated about its axis of symmetry generates a cone.

A worked model inside the lesson. The region between the lines y = 0 and y = 2 from x = 0 to x = 5 is a rectangle. Rotated about the x-axis it generates a cylinder of radius 2 and length 5, volume 20π. Rotated about the y-axis it generates a cylinder of radius 5 and height 2, volume 50π. Rotated about the line y = −1, which is parallel to a side and 1 unit away, it generates a tube of outer radius 3, inner radius 1 and length 5, volume π(9 − 1) × 5 = 40π. The same figure gives three different solids because the axis differs.

Misconceptions, and why they are wrong. "A cylinder's cross-section is a circle" is only true for cuts parallel to the base; a cut through the axis is a rectangle, and a tilted cut is an ellipse only when it crosses the curved surface without meeting either base. "A cube can be cut to give a circle" is false, since planes meet flat faces in straight segments. "Rotating a rectangle always gives the same cylinder" is false, as the 45π and 75π examples show; the axis matters. "A rotation produces a circle" confuses the path of one point with the solid swept by the whole figure. And a cross-section of a sphere that misses the center is still a circle, just smaller, never an ellipse.

Worked cylinder cuts: A cylinder has radius 4 cm and height 10 cm. Parallel to the base: a circle of radius 4 cm, area 16π ≈ 50.27 cm², the same at every height. Through the axis: a rectangle with width equal to the diameter, 8 cm, and height 10 cm, area 80 cm². Parallel to the axis at distance 2 cm from it: the width is the chord at distance 2 from the center of a circle of radius 4, which is 2√(4² − 2²) = 2√12 ≈ 6.93 cm, so the rectangle is 6.93 cm by 10 cm, area ≈ 69.28 cm². As the plane moves from the axis toward the surface the rectangle narrows to nothing.

Worked sphere and pyramid cuts: A sphere of radius 13 cm is cut by a plane 5 cm from its center. The cross-section is a circle of radius √(169 − 25) = 12 cm, area 144π ≈ 452.39 cm². Through the center the circle would have radius 13 cm and area 169π cm². A right square pyramid with base 10 cm and height 12 cm is cut parallel to the base 3 cm above it; the cut is 9 cm below the apex, so the square is scaled by 9/12 = 3/4: side 7.5 cm, area 56.25 cm², which is 100 × (3/4)².

Worked rotations of one triangle: A right triangle has legs 6 cm and 8 cm. Rotated about the 8 cm leg it sweeps a cone with height 8 cm and radius 6 cm, volume ⅓ × π × 36 × 8 = 96π ≈ 301.59 cm³. Rotated about the 6 cm leg it sweeps a cone with height 6 cm and radius 8 cm, volume ⅓ × π × 64 × 6 = 128π ≈ 402.12 cm³. Rotated about its hypotenuse (10 cm) it sweeps two cones sharing a circular base whose radius is the altitude to the hypotenuse, 6 × 8 ÷ 10 = 4.8 cm; their heights add to 10 cm, so the total volume is ⅓ × π × 4.8² × 10 = 76.8π ≈ 241.27 cm³.

Worked rectangle away from the axis: The rectangle 0 ≤ x ≤ 5, 1 ≤ y ≤ 3 is rotated about the x-axis. Points at y = 1 trace circles of radius 1 and points at y = 3 trace circles of radius 3, so the solid is a tube of length 5 with outer radius 3 and inner radius 1: volume π(9 − 1) × 5 = 40π ≈ 125.66 cubic units. The cross-section perpendicular to the axis is a ring, not a disc.

Cross-section catalog: For a cube, a cylinder, a cone, a sphere and a square pyramid, draw each solid and then sketch and name the cross-section for three cuts: parallel to the base, perpendicular to the base through the axis or apex, and one oblique cut. Record dimensions for one cut of each solid using these measurements: cube side 6 cm (parallel cut: square of side 6 cm), cylinder radius 4 cm and height 10 cm (axis cut: 8 cm by 10 cm rectangle), cone radius 5 cm and height 12 cm (axis cut: triangle base 10 cm, height 12 cm), sphere radius 13 cm cut 5 cm from the center (circle of radius 12 cm), pyramid base 10 cm and height 12 cm cut 3 cm above the base (square of side 7.5 cm).

Independent performance: Choose one two-dimensional figure drawn on coordinate axes (a rectangle, a right triangle or a right trapezoid) and rotate it about two different axes. The first axis is a side that meets its neighbor at a right angle: any side of the rectangle, a leg (not the hypotenuse) of the right triangle, or the leg of the right trapezoid that is perpendicular to its bases. The second axis is a line parallel to that side, outside the figure, on the far side of the chosen side, so that the chosen side lies between the axis and the rest of the figure. (A right triangle rotated about its hypotenuse, which gives two cones joined base to base, is taught above but is not one of the choices here.) For each rotation, name the solid generated, describe its cross-sections perpendicular to the axis, state its dimensions and compute its volume. Success criteria: each solid is named correctly (cylinder, tube, cone, frustum, or a cone or frustum with a cylindrical hole); the radius and height are read from the figure's position relative to the axis, not from the figure alone; the volume of the solid generated about the distant axis is computed as an outer solid minus an inner solid; one sentence explains why the two rotations of the same figure give different volumes; and one cross-section of each solid by a plane containing the axis is sketched and named.