Math 7 · 7.G.A.3 · G.GMD.B.4

Cross sections of solids

Cut straight through a solid, a block of tofu, a cucumber, a grapefruit. What flat shape does the knife leave behind, and what decides it?

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Prisms and pyramidsMath 6 Cylinders and conesMath 6 Three-dimensional shapesGrades K to 6

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Cavalieri's principleGeometry Conic sections from a double conePrecalculus Volumes with known cross sectionsCalculus 1

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When you reach for it

Cut a cucumber straight across and the knife leaves a round disc; cut it on a long slant and the disc stretches into an oval. The solid did not change. The direction of the cut did.

That flat face is a cross section, also called a plane section: the flat shape where a plane meets a solid. The plane is an ideal knife, perfectly flat and perfectly thin, and the cross section is everything it passes through, edge and inside together.

Reach for it when a question is about the inside of a solid rather than its outside (surface area): what shape a cut face will be, how a solid is built from thin slices, or which solid could have produced a given set of slices. The solids are the ones named in Three-dimensional shapes: prisms and pyramids, cylinders and cones, and Spheres.

Stand the solid on its base. A cut parallel to the base is level, like a slice across a loaf. A cut perpendicular to the base goes straight down. A cut on a slant is neither.

The procedure

Naming a cross section comes down to seeing which faces of the solid the cut crosses, and what each crossing looks like. Three facts do nearly all the work.

Three facts behind every cut

A flat cut through a flat face leaves a straight line. Two planes that meet, meet in a straight line (Lines and planes in space). So where the cut crosses one flat face, it leaves one straight segment, one side of the cross section. For a polyhedron, a solid whose faces are all flat polygons, this settles two things. The cross section is a polygon, so never a circle. And it has one side for each face the cut crosses, so no more sides than the solid has faces (Naming and classifying polygons).

Parallel faces give parallel sides. A plane meets two parallel planes in two parallel lines. So when the cut crosses two parallel faces, such as the top and bottom of a box, it leaves two parallel sides. This is what makes a cut a rectangle or a trapezoid rather than a four-sided shape with no parallel sides.

A cut parallel to the base copies the base. A prism is its base slid straight up, so every level holds the same shape at the same size. A pyramid is different: each of its points lies on a straight line from the tip, the apex, down to the base. A level cut crosses every one of those lines the same fraction of the way down, so it is a smaller copy of the base, a scale drawing of it, with

\text{scale factor} = \frac{\text{distance below the apex}}{\text{height}}

Halfway up, the scale factor is \displaystyle \frac{1}{2}: every length in the cut is half the matching length in the base.

Part 1: boxes and pyramids on a rectangle

A box is a prism three ways over, since any pair of opposite faces can serve as its bases. So a cut parallel to any face is a copy of that face. A cut straight down crosses the top and bottom in two level, parallel sides and the upright faces in two upright sides, which meet them at right angles: a rectangle as tall as the box. A cut on a slant can cross three, four, five or six faces, and the count decides the shape. A small cut off a corner crosses three faces, a triangle. A cut crossing four faces in two parallel pairs is a parallelogram, and a rectangle when the cut runs parallel to one of the box’s edges, as in the third panel. Sides of a cut that lie on hidden faces are drawn dashed, like the hidden edges.

Figure 1Three straight cuts through a box. Each cut face is a rectangle.

A pyramid on a rectangle, apex directly above the center of the base:

Parallel to the base: a smaller rectangle, scaled by the fraction above.
Straight down through the apex: it crosses the base and two side faces, three faces, so a triangle, as tall as the pyramid.
Straight down, parallel to a base edge, missing the apex: four faces, and a trapezoid (Example 3 shows why two sides are parallel).
On a slant, crossing the four side faces but not the base: a four-sided shape, generally with no special name.

Figure 2A square pyramid cut halfway up, through its tip, and in front of its tip.

A square pyramid has 5 faces, so no cut of it has more than 5 sides.

Part 2: other prisms and pyramids, and round solids

Nothing above used the base being a rectangle. For any prism standing straight up, a level cut is a copy of the base and a cut straight down is a rectangle as tall as the prism. For any pyramid, a level cut is a scaled copy of the base and a cut straight down through the apex is a triangle. The face count still caps the sides.

A round solid has a curved surface, and a flat cut through it can leave a curved edge. Each round solid follows the polyhedron it resembles.

A cylinder is a prism on a circle: the base circle slid straight up. A cut straight across is a circle the size of the ends. Its curved side is made of straight upright lines, so a cut lengthwise meets it in two of those lines and is a rectangle as tall as the cylinder. A cut on a slant that reaches neither end is an oval called an ellipse, exactly the diameter wide straight across and longer along the slant (Exercise 19 asks why). A slanted cut that runs into an end is an ellipse with a piece cut off by a straight side.

Figure 3A cylinder cut straight across, lengthwise, and on a slant.

A cone is a pyramid on a circle. A cut straight across is a smaller circle, scaled by the same fraction as for a pyramid. Its curved surface is made of straight lines from the apex to the base edge, so a cut straight down through the apex is a triangle. A cut on a slant that goes all the way round without reaching the base is again an ellipse.

Figure 4A cone cut straight across, through its tip, and on a slant.

A sphere gives only circles: it looks the same however it is turned, so turn it until the cut is level, and a level cut of a round solid is a circle (Solids of revolution). The cut through the center is the largest, with the sphere’s own radius; farther out the circles shrink, down to a single point where the plane only touches.

Figure 5Every cut of a sphere is a circle. The cut through the center is the largest.

Worked examples

Examples 1 to 4 go with Part 1; Examples 5 to 7 with Part 2.

Example 1

Three cuts through a block of tofu

Raisa has a block of tofu shaped like a rectangular prism, 8 cm long, 6 cm from front to back and 3 cm tall. Name the cross section, and give its dimensions and area, when she cuts it

(a) level, parallel to the base;

(b) straight down, parallel to the 8 cm by 3 cm front face;

Solution

(a) A cut parallel to the base of a prism is a copy of the base: a rectangle 8 cm by 6 cm, with area 8 \times 6 = 48 square centimeters, at whatever height she cuts.

(b) The cut crosses the top and bottom in two level sides 8 cm long and the ends in upright sides 3 cm tall, meeting at right angles: a rectangle 8 cm by 3 cm, with area 8 \times 3 = 24 square centimeters, a copy of the front face.

Example 2

Slanted cuts through the tofu

Raisa takes a second block the same size. Name the cross section when

(a) she cuts from the top front edge straight through to the bottom back edge, keeping the knife level from left to right;

(b) she makes a small cut that takes one top corner off the block.

Solution

(a) The cut contains two whole edges, the top front and the bottom back, both 8 cm long: two sides. On each end it runs corner to corner along a diagonal of the 6 cm by 3 cm end: two more. Each 8 cm edge is at right angles to the end faces, so to the diagonals lying in them. Every corner is a right angle, and the cross section is a rectangle (third panel of Figure 1).

One side is 8 cm. The other is a diagonal of the end, the longest side of a right triangle with shorter sides 6 cm and 3 cm, so it is more than 6 cm. This slanted rectangle is larger than the 8 cm by 6 cm base, the largest level or upright cut.

(b) A small cut near a corner enters only the three faces that meet there: the top, the front and the end. Three faces give three sides, a triangle. If the cut meets the three edges at the corner all at the same distance from it, each side of the triangle is the long side of a right triangle whose two shorter sides are both that distance. Those three right triangles are copies of each other, so the triangle is equilateral.

Example 3

Slicing a pyramid

Oriana has a bar of soap shaped like a square pyramid: the base is 10 cm by 10 cm, the height is 12 cm, and the apex is directly above the center of the base. Name the cross section, giving its size where the numbers allow, when she cuts it

(a) level, halfway up;

(b) straight down through the apex, parallel to one edge of the base;

Solution

Figure 2 shows all three cuts, in this order.

(a) A level cut of a pyramid is a smaller copy of the base. Halfway up it is 6 cm below the apex, so the scale factor is \displaystyle \frac{6}{12} = \frac{1}{2}. The cross section is a square with side \displaystyle 10 \times \frac{1}{2} = 5 cm and area 5 \times 5 = 25 square centimeters. The base has area 100, so halving both lengths left a quarter of the area (Scale factor, area and volume).

(b) The cut crosses the base and the left and right side faces, which meet at the apex: a triangle. Its bottom side runs across the base through the center, parallel to an edge, so it is 10 cm long. The apex lies in the cut directly above the middle of that side, so the triangle’s height is the pyramid’s, 12 cm, and its slanted sides are mirror images: it is isosceles, with area \displaystyle \frac{1}{2} \times 10 \times 12 = 60 square centimeters.

(c) Now the cut misses the apex. It crosses the base, the left and right faces, and the front face: four sides. The bottom side lies in the base, parallel to the front edge. The top side lies in the front face, which contains that edge, and a plane parallel to an edge meets a plane containing it in a line parallel to the edge. Both sides are parallel to the front edge, so to each other: the cross section is a trapezoid, wider at the bottom, and isosceles by the same mirror symmetry as in (b).

Example 4

What one cut can do to a cube

Sebastien says one straight cut through a cube can leave a triangle, a square, a rectangle that is not a square, a pentagon or a hexagon, but never a heptagon, an octagon or a circle. Is he right? Describe a cut for each shape he names.

Solution

The limits first. A cube has 6 faces and each face crossed gives one side, so at most 6 sides: no heptagon (7) and no octagon (8). Every side lies in a flat face, so every side is straight, and a circle is impossible.

Each shape he names occurs.

Square: cut parallel to a face, giving a copy of the face.
Rectangle that is not a square: cut straight down through two opposite upright edges. Two sides are those edges, as long as a side of the cube; the other two are diagonals of the top and bottom squares, which are longer.
Triangle: cut off a corner, crossing the three faces that meet there.
Pentagon: a slanted cut that crosses the top, the bottom and three of the four upright faces, missing only one face.
Hexagon: a cut through the midpoints of the six edges that touch neither of two opposite corners. It crosses all six faces. Each side joins the midpoints of two edges meeting at a corner of a face, so all six sides are equal, and the cube’s symmetry makes the angles equal: a regular hexagon.

Figure 6One straight cut through a cube can cross three, four, five or all six faces, and the cut face has that many sides.

So Sebastien is right, though his list is not complete: a cube can also be cut into a trapezoid, or a parallelogram that is not a rectangle.

Example 5

Other prisms and pyramids

Marguerite has two solids: a triangular prism 10 cm tall, standing on a base triangle with sides 5 cm, 5 cm and 6 cm; and a pyramid whose base is a regular hexagon with sides 6 cm, its apex above the center. For each, name the cross section of a level cut halfway up and of a cut straight down, and give the greatest number of sides any cross section can have.

Figure 7The rules for the box and the square pyramid hold for every prism and every pyramid.

Solution

The triangular prism. A level cut is a copy of the base: a triangle with sides 5 cm, 5 cm and 6 cm. A cut straight down crosses the top and bottom triangles in two level sides and two of the three upright faces in two upright sides 10 cm tall: a rectangle 10 cm tall. The prism has 2 + 3 = 5 faces, so at most 5 sides, and 5 really happens: a slanted cut that leaves two bottom corners and one top corner on one side of it, the top corner above one of those two, crosses both triangles and all three upright faces, and leaves a pentagon.

The hexagonal pyramid. A level cut halfway up is a copy of the base at scale factor \displaystyle \frac{1}{2}: a regular hexagon with sides \displaystyle 6 \times \frac{1}{2} = 3 cm. A cut straight down through the apex crosses the base and two side faces: a triangle. The pyramid has 1 + 6 = 7 faces, so no cross section has more than 7 sides.

Example 6

A cucumber cut four ways

Nikolaus has a cucumber that is very nearly a cylinder, 4 cm across and 20 cm long. Name the cross section, and give its size where the numbers allow, when he cuts it

(a) straight across;

(b) lengthwise, through the middle;

(d) on a long slant that reaches neither end.

Solution

(a) A copy of the end: a circle 4 cm across, so with radius \displaystyle \frac{4}{2} = 2 cm.

(b) The cut meets the curved side along two full-length straight lines, a diameter apart, and each end along a diameter. So it is a rectangle 20 cm by 4 cm, with area 20 \times 4 = 80 square centimeters.

(c) Still a rectangle 20 cm long, but narrower than 4 cm: it crosses each end along a line that misses the center, shorter than the diameter. A cut that only grazes the skin leaves a single line.

(d) An ellipse. Measured straight across the cucumber it is exactly 4 cm wide, the diameter; along the slant it is longer, and the longer the slant, the longer the oval.

Example 7

A cone of foam and a grapefruit

Ptolemy has a cone of craft foam with base radius 3 cm and height 8 cm, and a grapefruit that is very nearly a sphere of radius 5 cm.

(a) Name the cross section of the cone cut level, halfway up, and give its radius.

(b) Name the cross section of the cone cut straight down through the apex, and give its area.

Solution

(a) As for a pyramid, halfway up the scale factor is \displaystyle \frac{1}{2}, so the cross section is a circle of radius \displaystyle 3 \times \frac{1}{2} = 1.5 cm.

(b) The cut meets the base along a diameter, 2 \times 3 = 6 cm long, and the curved surface along two straight lines from the apex to the ends of that diameter: an isosceles triangle with base 6 cm and height 8 cm, and area \displaystyle \frac{1}{2} \times 6 \times 8 = 24 square centimeters.

(c) Every cut is a circle. The largest goes through the center with the sphere’s radius, 5 cm, so it is 10 cm across. The farther a cut is from the center, the smaller its circle.

Boundaries

The cut is one flat plane. A curved cut, or two cuts at an angle, is not a cross section.

A plane that only touches the solid leaves no slice through it. It leaves a point at a vertex, a segment along an edge, or the face itself against a face. A plane that misses has no cross section.

One side per face needs a solid without dents. Every solid here is convex: a straight line between any two of its points stays inside it. A solid with a dent, such as an L-shaped block, can have a face that one cut crosses twice, and a cut through it can fall in two separate pieces.

Straight down gives a rectangle only when the solid stands straight. In a leaning prism or cylinder, whose side edges are not at right angles to the base, a cut parallel to the side edges is a parallelogram, in general not a rectangle. A level cut is still a copy of the base.

The scale factor rule needs a level cut, measured from the apex. It holds for every pyramid and cone, leaning or not, but only for cuts parallel to the base, with the distance measured down from the apex. The triangles and trapezoids above are isosceles only because the apex is directly above the center of the base.

The ellipse needs the slant cut to go all the way round. A cut of a cone as steep as the cone’s side, or steeper, never closes up: its curved edge runs down to the base, and those curves are studied in Conic sections from a double cone.

Where this goes next

The slices of a solid hold its volume. Cavalieri’s principle says that two solids whose cross sections at every height have equal areas have equal volumes, which is why a leaning stack of coins holds as much as an upright one. The soap pyramid’s halfway square, with a quarter of the base’s area, is the first clue to why a pyramid holds one third of its prism: its slices shrink fast as they rise. In calculus the idea becomes volumes from known cross sections, adding up the areas of infinitely thin slices.

The cone’s cuts, level, slanted, parallel to the side and steeper, give the four conic sections, and Conic sections from a double cone shows all four coming from one cone. The ellipse later gets a definition that never mentions a cone.

Exercises

Exercises 1 to 12 go with Part 1, Exercises 13 to 23 with Part 2; the reasoning group draws on both. In every pyramid and cone below, the apex is directly above the center of the base.

Part 1: boxes

A block of tofu is a rectangular prism 10 cm long, 7 cm from front to back and 4 cm tall. Name the cross section, and give its dimensions and area, when it is cut:

level, parallel to the 10 cm by 7 cm base.
straight down, parallel to the 10 cm by 4 cm front face.
straight down, parallel to the 7 cm by 4 cm end.
A cube with edges 6 cm long is cut parallel to one face. Name the cross section and give its area.
The same cube is cut straight down through two opposite upright edges, along a diagonal of the top. Name the cross section, and say how its sides compare with 6 cm.

Part 1: pyramids

A bar of soap is a square pyramid with base 8 cm by 8 cm and height 10 cm.

It is cut level, halfway up. Name the cross section, and give its side and area.
It is cut level, 2.5 cm below the apex. Name the cross section, and give its side and area.
It is cut straight down through the apex, parallel to an edge of the base. Name the cross section and give its area.
A pyramid has a rectangular base 12 cm by 6 cm and height 9 cm. It is cut straight down through the apex, parallel to the 12 cm edges. Name the cross section and give its area.
Describe a cut of a square pyramid whose cross section is a trapezoid, and explain why that cross section has two parallel sides.

Part 1: cubes

A cut passes through the three corners of a cube that are joined by an edge to one particular corner. Name the cross section as exactly as you can, and explain.
Can one straight cut through a cube leave a heptagon or an octagon? Explain.

Part 2: other prisms and pyramids

A triangular prism stands on a base triangle with sides 3 cm, 4 cm and 5 cm. It is cut level, parallel to the base. Name the cross section and give its perimeter.
The same prism is 10 cm tall. It is cut straight down, and the cut crosses the base triangle along a line 2.5 cm long. Name the cross section and give its area.
A pyramid on a regular hexagon with sides 6 cm is cut level, halfway up. Name the cross section and give the length of its sides.
A pentagonal prism has a pentagon at each end. Using one side per face crossed, what is the greatest number of sides a cross section of it can have?

Part 2: round solids

A cucumber is very nearly a cylinder 3 cm across and 18 cm long.

It is cut straight across. Name the cross section and give its radius.
It is cut lengthwise through the middle. Name the cross section and give its area.
It is cut on a long slant that reaches neither end. Name the cross section, and compare its narrowest width with 3 cm.

A cone of craft foam has base radius 4 cm and height 12 cm.

It is cut level, halfway up. Name the cross section and give its radius.
It is cut straight down through the apex. Name the cross section and give its area.
A grapefruit is very nearly a sphere 12 cm across. Name the cross section of a cut through its center and give its radius. Is a cut 3 cm from the center larger or smaller?
Among a cylinder, a cone and a sphere, which solid could give (a) circles, rectangles and ellipses, (b) circles, triangles and ellipses, (c) only circles?

Reasoning

Explain why every cut of a prism parallel to its base is a copy of the base, the same size, while every such cut of a pyramid is a smaller copy.
Explain why no cut of a cube is a circle, while a cut of a cylinder can be.

Answers

A rectangle 10 cm by 7 cm, with area 70 square centimeters. A level cut copies the base.
A rectangle 10 cm by 4 cm, with area 40 square centimeters, a copy of the front face.
A rectangle 7 cm by 4 cm, with area 28 square centimeters, a copy of the end.
A square 6 cm by 6 cm, with area 36 square centimeters, a copy of the face.
A rectangle that is not a square. Two of its sides are upright edges of the cube, 6 cm long; the other two are diagonals of the top and bottom faces. Each diagonal is the longest side of a right triangle whose other sides are 6 cm, so it is longer than 6 cm. The top and bottom sides are parallel and the upright sides meet them at right angles.
A square with side 4 cm and area 16 square centimeters. The cut is 5 cm below the apex, so the scale factor is \displaystyle \frac{5}{10} = \frac{1}{2}, and \displaystyle 8 \times \frac{1}{2} = 4.
A square with side 2 cm and area 4 square centimeters. The scale factor is \displaystyle \frac{2.5}{10} = \frac{1}{4}, and \displaystyle 8 \times \frac{1}{4} = 2.
An isosceles triangle with base 8 cm and height 10 cm, with area \displaystyle \frac{1}{2} \times 8 \times 10 = 40 square centimeters.
An isosceles triangle with base 12 cm and height 9 cm, with area \displaystyle \frac{1}{2} \times 12 \times 9 = 54 square centimeters.
Cut straight down, parallel to one edge of the base, between the apex and that edge. The bottom side lies in the base, parallel to the edge. The top side lies in the face containing that edge, and a plane parallel to an edge meets a plane containing the edge in a line parallel to it. Both are parallel to the edge, so they are parallel to each other.
An equilateral triangle. The cut crosses three faces, so it is a triangle, and each side joins two corners of one square face across its diagonal. All the faces are equal squares, so the three diagonals are equal.
No. Each side of a cross section lies in a different face, and a cube has only 6 faces, so a cross section has at most 6 sides.
A triangle with sides 3 cm, 4 cm and 5 cm, a copy of the base, with perimeter 3 + 4 + 5 = 12 cm.
A rectangle 2.5 cm by 10 cm, with area 25 square centimeters.
A regular hexagon with sides 3 cm. The scale factor halfway up is \displaystyle \frac{1}{2}.
7. The prism has 5 upright faces and 2 ends, 7 faces in all, and a cross section has at most one side per face.
A circle of radius 1.5 cm, a copy of the end.
A rectangle 18 cm by 3 cm, with area 54 square centimeters.
An ellipse, an oval. The cut crosses the cucumber’s central line at one point, and the level line through that point in the cut is a diameter of the round slice there, so the narrowest width is exactly 3 cm. Along the slant the oval is longer than 3 cm.
A circle of radius 2 cm. The scale factor halfway up is \displaystyle \frac{1}{2}.
An isosceles triangle with base 8 cm, the diameter, and height 12 cm, with area \displaystyle \frac{1}{2} \times 8 \times 12 = 48 square centimeters.
A circle of radius 6 cm. The cut 3 cm from the center is a smaller circle: only the cut through the center has the sphere’s own radius.
(a) The cylinder: across, lengthwise and on a slant. (b) The cone: across, through the apex and on a slant. (c) The sphere, whose every cut is a circle. A cone gives no rectangle, since its sides close in to the apex, and a cylinder gives no triangle, since its sides never meet.
A prism is its base slid straight along, so every level holds the same shape at the same size. In a pyramid every point lies on a line from the apex to the base, and a level cut crosses all those lines the same fraction of the way down. Every length in the cut is that fraction of the matching length in the base, which is less than 1 above the base, so the cut is a smaller copy.
Every face of a cube is flat, and a flat cut meets a flat face in a straight line, so every side of a cut of a cube is straight and no edge can be curved. A cylinder has a curved surface, and a level cut meets it in a circle, the same as its end.