Math by First Principles

Cross-Sections and Solids of Revolution

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Learning goals

  • Identify true plane sections from the solid and the cut's stated orientation
  • Distinguish circular sections from elliptical sections and misleading perspective drawings
  • Describe cylinders, cones, and balls generated by rotating familiar plane regions
  • Compare dimensions produced by different named axes of rotation
  • Reconstruct a cutting plane or rotation axis from geometric constraints

A slice has its own plane

A cross-section is the intersection of a plane with a solid. Picture an ideal knife with no thickness. We study the entire flat region it cuts, rather than a chunk with thickness. A circular section is a disk, bounded by a circle; shape names below describe these flat sections.

For a box 6 cm by 8 cm by 5 cm, a cut parallel to the 6 cm by 8 cm base makes another 6 cm by 8 cm rectangle. The same upright edges meet both planes, so corresponding corners move the same distance. A prism’s interior base-parallel sections are congruent to its base.

Name the reference before naming the orientation. A plane can be parallel to the base, perpendicular to the base, or oblique (neither). “Vertical” alone means little unless the solid’s position is fixed. Unless stated otherwise, our planes pass through the interior and make a section with positive area. A plane that only touches a vertex is a boundary case, not one of these ordinary slices.

Worked example 1 A diagonal cut can still make a rectangle

Problem. A right rectangular box has a 6 cm by 8 cm base and height 5 cm. Cut it with the plane perpendicular to the base that contains one base diagonal. Name the true section and find its dimensions and area.

Solution. The cut follows the diagonal across the base and rises along the perpendicular upright edges. Its true shape is a rectangle. Its width is the base diagonal, not either base side.

d2=62+82d2=100d=10 cm.\begin{aligned} d^2&=6^2+8^2\\ d^2&=100\\ d&=10\text{ cm}. \end{aligned}

The section is 10 cm by 5 cm. Its area is

A=50 cm2.A=50\text{ cm}^2.
A box cut by a plane perpendicular to its base through a base diagonal, with its true rectangular sectionIn the left box, the plane follows a base diagonal and the upright edges through its endpoints. Hidden box edges are dashed. The right panel shows the section directly, with its diagonal-width and perpendicular-height dimensions.3D viewTrue 2D685105
The blue section is projected on the box; the dashed red rectangle is its true shape. All dimensions are in centimeters. Hidden box edges are dashed.

Check your understanding

An interior cut through a triangular prism is parallel to its triangular bases. What is the true section?

Flat faces give polygonal sections

When a plane crosses a flat face, their intersection is a straight segment. Following those segments around the cut gives a polygon. Count the faces actually crossed; do not count all the faces of the solid.

For a square pyramid, a cut halfway between the base and apex, parallel to the base, makes a smaller square. Moving that plane toward the apex makes smaller squares. A plane through the apex and a base diagonal makes an isosceles triangle in a right square pyramid. A perpendicular plane missing the apex need not make that triangle.

Predict each section before reading its right-hand panel.

A square pyramid with a selected plane section and its true two-dimensional shapeThe left pyramid has a four-corner base and an apex. Dashed strokes identify hidden edges. The highlighted plane section is also drawn separately at right, viewed directly in the cutting plane.3D viewTrue 2Dsquare
A base-parallel cut of this square pyramid gives a smaller square. The dashed red panel faces the cutting plane directly.
A square pyramid with a selected plane section and its true two-dimensional shapeThe left pyramid has a four-corner base and an apex. Dashed strokes identify hidden edges. The highlighted plane section is also drawn separately at right, viewed directly in the cutting plane.3D viewTrue 2Dtriangle
A cut through the apex and opposite base vertices gives a triangle. The dashed red panel shows its true shape.

A triangular prism, a square pyramid, and a cube can have different sections when you tilt the plane. “Oblique cut” is an orientation description, not a shape name. Locate the cut’s intersections with the faces to determine the polygon.

Check your understanding

A plane cuts the interior of a cube, crosses exactly five faces, contains no cube edge, and passes through no vertex. How many sides does its section have?

Curved solids and the true shape

A circular cut can look oval in a picture. A horizontal slice through a right circular cylinder of radius 4 cm has radius 4 cm in its own plane. Drawing that circle slanting away from you compresses its appearance; it does not change the section into an ellipse.

A right cylinder with a selected plane section and its true two-dimensional shapeThe left perspective view shows two circular end disks and joining side lines, with the selected interior section highlighted. Rear end arcs are dashed. The right panel faces the cutting plane directly, with a dashed outline distinguishing the true shape from the projected shape.3D viewTrue 2Dcircle
The base-parallel cut is circular. The blue oval is its projection; the dashed red circle is the true section.

Here ellipse names the noncircular oval produced by the cut described next. Its exact definition is for a later course. An oblique plane through a right circular cylinder makes an elliptical section when its entire boundary lies on the curved side and it misses both end disks. Looking straight at that plane still shows an oval.

A right cylinder with a selected plane section and its true two-dimensional shapeThe left perspective view shows two circular end disks and joining side lines, with the selected interior section highlighted. Rear end arcs are dashed. The right panel faces the cutting plane directly, with a dashed outline distinguishing the true shape from the projected shape.3D viewTrue 2Dellipse
This tilted cut misses both ends. Its true section is a noncircular ellipse, shown dashed at right.

Worked example 2 Choose the plane before naming the section

Problem. Classify these sections. All solids are right and circular where appropriate, and every cut passes through the interior.

(a) A cylinder is cut by a plane containing its axis (the line through the centers of its bases).

(b) A cone is cut parallel to its base, between its base and apex.

(c) A cone is cut by a plane containing its axis.

Solution. (a) The plane meets the two end disks in diameters and the curved side in two upright segments. The section is a rectangle, with width equal to the cylinder’s diameter and height equal to its height.

A right cylinder with a selected plane section and its true two-dimensional shapeThe left perspective view shows two circular end disks and joining side lines, with the selected interior section highlighted. Rear end arcs are dashed. The right panel faces the cutting plane directly, with a dashed outline distinguishing the true shape from the projected shape.3D viewTrue 2Drectangle
A plane containing the cylinder axis cuts a rectangle. Its width is a diameter.

(b) Each direction around the cone gives the same radius at that level. The section is a circle, smaller than the base circle.

A cone cut parallel to its base or through its axis, with the true section beside itThe left perspective cone has an apex above a circular base. The highlighted section follows either an interior base-parallel plane or a plane through the axis. The right dashed outline shows the same section viewed directly in its own plane.3D viewTrue 2Dcircle
A cone cut parallel to its base has a circular section, despite its oval projection.

(c) The plane contains the apex and a base diameter. The section is an isosceles triangle, not a right triangle formed by just one radius.

A cone cut parallel to its base or through its axis, with the true section beside itThe left perspective cone has an apex above a circular base. The highlighted section follows either an interior base-parallel plane or a plane through the axis. The right dashed outline shows the same section viewed directly in its own plane.3D viewTrue 2Dtriangle
The whole axial section includes two radius-halves. Its true shape is an isosceles triangle.

An oblique plane can also make an elliptical section of a cone if it misses the apex and base and makes a closed cut entirely around the curved side. We are selecting these familiar cuts, not classifying every possible cut of a cone.

A sphere is the surface consisting of points a fixed distance from its center. A ball includes the region inside it. A plane cutting through a sphere meets the surface in a circle; the same plane cuts the ball in the disk inside that circle. Even a tilted slice is circular, although its perspective picture may look oval.

A sphere with a plane section appearing oval in projection and shown as a true circle beside itThe left circular outline represents the sphere in projection. The highlighted off-center plane section looks oval because it faces away. The right dashed circle shows that section viewed directly in its own plane.3D viewTrue 2Dcircle
A plane through the sphere interior meets its surface in a circle. The ball section is the enclosed disk.

The section becomes smaller as its plane moves farther from the center. For example, in a sphere of radius 5 cm, a plane 3 cm from the center gives a section radius of 4 cm. The sphere center, section center, and a point on the section form a right triangle. Its hypotenuse is a sphere radius.

32+r2=52r2=16r=4 cm.\begin{aligned} 3^2+r^2&=5^2\\ r^2&=16\\ r&=4\text{ cm}. \end{aligned}
The sphere center O, section center M, and section boundary point P form a right triangle, beside the true circular sectionIn the left center view, O is the sphere center, M is the perpendicular foot on the cutting plane, and P is on the sphere boundary in that plane. OP is a sphere radius, OM is the perpendicular center distance, and MP is the section radius. A right-angle mark is at M. The supplied lengths and the requested length are labeled; the true section is drawn at right.Center viewOMP534True sectioncircle
OM is perpendicular to the cutting plane, MP is a section radius, and OP is a sphere radius. Lengths are in centimeters; the center view shows the right triangle without perspective.

This uses perpendicular distance to the plane. The largest circle occurs when the plane passes through the center. A tangent plane meets the sphere in only one point, and a plane farther away misses it.

Check your understanding

A plane parallel to a cylinder's base produces a section drawn as a narrow oval in perspective. What is its true shape?

Turning a region about a named axis

A solid of revolution is swept out when a filled plane region turns a full revolution about a line in its plane. That fixed line is the axis of rotation. A point 3 cm from the axis traces a circle of radius 3 cm; a point on the axis stays fixed. Following the points of the entire region fills the solid.

Worked example 3 The side on the axis becomes the height

Problem. Rectangle ABCD has side AB of 7 cm and perpendicular side AD of 3 cm. Compare the solids formed by a full turn about each named side.

(a) Rotate the filled rectangle about AB.

(b) Rotate it about AD.

Solution. (a) Side AB stays on the axis, so it becomes the height of a right circular cylinder. The far side is 3 cm from that axis. The cylinder has height 7 cm and radius 3 cm.

A rectangle with named sides and the cylinder generated about the stated sideA is the lower left corner, B the upper left, C the upper right, and D the lower right. AB and AD have the given length labels. The named fixed side is the rotation axis. At right, a dashed cylinder outline shows the result and its height and radius labels.Rectangle ABCDAbout ABABCD73h 7, r 3
Rotate about AB. The fixed side is 7 cm long and the radius swept out is 3 cm. The cylinder is a schematic perspective view.

(b) With AD fixed, the height is 3 cm. The far side is now 7 cm from the axis, so the radius is 7 cm. This cylinder is wider and shorter than the first. The distance from the axis is a radius, not a diameter.

A rectangle with named sides and the cylinder generated about the stated sideA is the lower left corner, B the upper left, C the upper right, and D the lower right. AB and AD have the given length labels. The named fixed side is the rotation axis. At right, a dashed cylinder outline shows the result and its height and radius labels.Rectangle ABCDAbout ADABCD73h 3, r 7
Rotate about AD. The fixed side is 3 cm long and the swept radius is 7 cm. The named axis, not the page orientation, sets height and radius.

Check your understanding

A filled rectangle is 4 cm by 9 cm. It turns about one of its 9 cm sides. Which cylinder results?

Worked example 4 A tapering region and a round region

Problem. Predict the solids and dimensions in these two full rotations.

(a) Right triangle ABC has right angle A, leg AB of 12 cm, and leg AC of 5 cm. Rotate its filled region about AB.

(b) Rotate a filled semicircular region of radius 3 cm about its diameter.

Solution. (a) AB stays fixed. AC sweeps the base disk of radius 5 cm. The hypotenuse sweeps the curved side, narrowing to B. The result is a right circular cone of height 12 cm and base radius 5 cm. Turning about AC instead would give height 5 cm and radius 12 cm.

A right triangle with right angle A and axis AB, beside the resulting right coneA is the right-angle vertex. AB is the fixed axis leg and AC is the perpendicular leg reaching away from it. Their lengths are labeled. At right, the cone apex lies at B and the base radius is swept by AC; a dashed rear base arc shows depth.Right triangle ABCAbout ABABC125h 12, r 5
Leg AB is the 12 cm axis. The perpendicular 5 cm leg sweeps the cone base radius; both panels are schematic.

(b) The semicircular arc sweeps the sphere of radius 3 cm. The filled region sweeps the whole ball of radius 3 cm, not half a ball. Each segment from the diameter to the arc, perpendicular to the diameter, sweeps a disk during the turn.

A filled semicircular region turns about its diameter and sweeps out a ballThe left half-disk lies above its horizontal diameter, which is extended as a dashed rotation axis. The right ball has a dashed sphere boundary and a separate dashed equatorial circle in perspective.Filled half-diskAfter a full turndiameter axisball
The filled region turns about its diameter and fills a ball. The outer arc sweeps its sphere boundary. The sphere is drawn in perspective.

Work backward from a section

Sections can constrain a solid without showing every part of it. Be precise about what they determine: one circular slice alone could belong to a cylinder, cone, or ball.

Worked example 5 Recover a generating rectangle

Problem. A right circular cylinder has a base-parallel section of area 25π25\pi square centimeters. Its section through the axis is a rectangle 10 cm wide and 8 cm high. A filled rectangle will generate this cylinder by turning about one of its sides. Give the rectangle’s dimensions, name the side to turn about, and check both sections.

Solution. The circular section fixes the radius.

πr2=25πr=5 cm.\begin{aligned} \pi r^2&=25\pi\\ r&=5\text{ cm}. \end{aligned}

The axis section spans two radii, so its 10 cm width agrees. Its 8 cm height is the cylinder’s height.

Use a 5 cm by 8 cm rectangle, rotating about an 8 cm side. The other side sweeps radius 5 cm. The circular sections have area 25π25\pi square centimeters, and the axial rectangle has width 10 cm and height 8 cm.

A rectangle with named sides and the cylinder generated about the stated sideA is the lower left corner, B the upper left, C the upper right, and D the lower right. AB and AD have the given length labels. The named fixed side is the rotation axis. At right, a dashed cylinder outline shows the result and its height and radius labels.Rectangle ABCDAbout ABABCD85h 8, r 5
Choose AB as the 8 cm axis and AD as the 5 cm radius. The generated axial section is twice as wide as the original rectangle.

Check your understanding

A right cone is generated by turning a filled right triangle about one leg. The cone has height 6 cm and radius 11 cm. Which choice works?

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A bit of history (optional)

An ordinary X-ray puts overlapping structures into one picture. In the 1960s, the engineer Godfrey Hounsfield worked on a different approach: collect X-ray measurements from many directions and use a computer to reconstruct images of separate cross-sections.

His experimental brain scanner was installed at Atkinson Morley’s Hospital in London. The first patient was examined on October 1, 1971. The result let doctors inspect an internal slice without physically cutting the patient. Hounsfield and Allan Cormack shared the 1979 Nobel Prize in Physiology or Medicine for developing computer assisted tomography. Cormack had independently developed mathematical work behind this new kind of imaging.

The Nobel Foundation describes the development of the scanner and Hounsfield’s work.

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