8.2 Cross-Sections & Solids of Revolution
Key Takeaways
- A cross-section parallel to a base reproduces the base shape: a cylinder sliced parallel to its base gives a circle congruent to the base.
- Rotating a rectangle about a side gives a cylinder; a right triangle about a leg gives a cone; a semicircle about its diameter gives a sphere.
- A 6-by-4 rectangle spun about the side of length 6 forms a cylinder with radius 4 and height 6, volume 96π.
- Cavalieri's principle: solids with equal heights and equal-area cross-sections at every level have equal volume.
- Standard GEO-G.GMD.4 tests moving both directions between 2D figures and 3D solids.
Slicing and Spinning: Two Ways 2D Meets 3D
Standard GEO-G.GMD.4 asks you to move between two and three dimensions in two directions: identify the cross-section you get when a plane slices a solid, and identify the solid of revolution you get when a flat figure spins around a line. These questions appear as quick multiple-choice items and inside Part II–IV modeling problems, so both directions are worth practicing.
Cross-Sections: Slicing a Solid
A cross-section is the two-dimensional shape formed where a plane cuts through a solid. The shape depends on the angle of the cut relative to the solid.
| Solid | Cut parallel to base | Cut perpendicular to base |
|---|---|---|
| Cylinder | circle (congruent to base) | rectangle |
| Rectangular prism | rectangle congruent to base | rectangle |
| Cone | circle (smaller than base) | triangle (through apex) |
| Square pyramid | square similar to base | triangle (through apex) |
| Sphere | circle (any cut) | circle |
Two ideas cover most exam questions. First, a slice parallel to the base reproduces the base shape: a cylinder cut parallel to its circular base yields a circle congruent to the base, and a rectangular prism cut parallel to its base yields a rectangle congruent to the base.
Second, a slanted or angled cut can produce a new shape — a cone cut parallel to its slant side produces a parabola, and an angled cut through a cylinder produces an ellipse. A sphere is the friendly case: every flat cut is a circle, and the largest possible one, through the center, is called a great circle. A cut through the apex of a cone or pyramid — perpendicular to the base — produces a triangle rather than a copy of the base.
Solids of Revolution: Spinning a Figure
Rotate a two-dimensional figure a full 360° about a straight line (the axis of rotation) and it sweeps out a solid.
- Rotate a rectangle about one side → a cylinder (the side on the axis is the height; the perpendicular side is the radius).
- Rotate a right triangle about a leg → a cone (the leg on the axis is the height; the other leg is the radius).
- Rotate a semicircle about its diameter → a sphere.
Worked example — cylinder. A 6-by-4 rectangle is rotated about the side of length 6. The side on the axis becomes the height (6) and the perpendicular side becomes the radius (4). So V = πr²h = π(4²)(6) = 96π. Had it spun about the side of length 4, the radius and height would swap, giving π(6²)(4) = 144π — a different solid, so read which side is the axis.
Worked example — cone. A right triangle with legs 5 and 12 is rotated about the leg of length 5. That leg is the height (5) and the other leg is the radius (12): V = ⅓π(12²)(5) = 240π. Rotating about the leg of 12 instead would give radius 5, height 12 — a completely different cone with volume ⅓π(5²)(12) = 100π. Always identify which segment lies on the axis: that length is the height, and the perpendicular distance from the axis to the farthest point is the radius.
Worked example — sphere. Rotating a semicircle of radius 6 about its diameter sweeps out a sphere of radius 6, so V = (4/3)π(6³) = (4/3)π(216) = 288π. If instead you rotate a full circle about a line beside it you get a torus (a donut), which is beyond the exam but shows how the axis choice controls the solid.
Cavalieri's Principle — Why the Formulas Work
Cavalieri's principle (GEO-G.GMD.1) states that if two solids have the same height and every parallel cross-section at the same level has equal area, then the two solids have equal volume. This is why an oblique (leaning) cylinder has the same volume as an upright cylinder with the same base and height, and why a slanted stack of coins holds the same volume as a straight stack. It also justifies V = Bh and V = ⅓Bh for oblique prisms, cylinders, cones, and pyramids: leaning a solid over does not change the area of any horizontal slice, so it does not change the volume.
Connecting the Two Directions
The two skills are inverses. Rotating a right triangle creates a cone; slicing that cone perpendicular through its apex returns a triangle. Being able to picture the slice or the spin lets you set up a volume formula even when a figure is described only in words.
One more pattern worth knowing: horizontal cross-sections of a pyramid or cone taken between the base and the apex are similar to the base but smaller, shrinking to a single point at the apex. That shrinking-similar-slice picture is exactly the informal dissection argument behind the ⅓ factor in V = ⅓Bh, tying this section back to the reference-sheet formulas. On the multiple-choice items, sketch the solid, mark the cutting plane or the axis, and name the resulting shape before checking the answer choices — a two-second drawing prevents the most common errors.
A plane cuts a right circular cylinder in a plane parallel to its base. What is the shape of the cross-section?
A right triangle with legs 5 and 12 is rotated 360° about the leg of length 5. What is the volume of the solid formed?
Two solids have the same height, and at every level their parallel cross-sections have equal area. Which conclusion follows?