Prisms and pyramids
A tissue box, a glass prism from a science lab and a pyramid tea bag are all built from flat faces. What sorts them into prisms and pyramids, and how does one polygon, the base, decide how many faces, edges and vertices the whole solid has?
Where you are 7 in Three-dimensional shapes
Why this exists
Polyhedra, solids made only of flat polygons, come in endless shapes, and counting the parts of each new one from scratch is slow. But two families turn up again and again, in boxes, buildings, packaging and molds. In one, the same polygon sits at both ends with flat walls joining them. In the other, one polygon sits at the bottom and triangles rise from its sides to a point. Naming these families, prisms and pyramids, lets a few words describe a whole solid, and it turns counting into a rule: know the base, and you know how many faces, edges and vertices there are.
Definition
A prism is a polyhedron with two faces, its bases, that are exact copies of each other (the same shape and size, matching corner to corner) and lie in parallel planes: flat, facing each other, the same distance apart everywhere, like the floor and ceiling of a room. Every other face joins a side of one base to the matching side of the other and is a parallelogram. These are the lateral faces (“lateral” means “at the side”). The edges joining matching corners of the two bases are the lateral edges.
A pyramid is a polyhedron with one polygon face, its base, and one point off the base, its apex. Every other face is a triangle made by one side of the base and the apex. These triangles are its lateral faces, and the edges from the apex to the corners of the base are its lateral edges.
Names. A prism or pyramid is named by the shape of its base: triangular, rectangular, pentagonal, hexagonal, octagonal and so on, using the polygon names of Naming and classifying polygons. A cube is a square prism whose faces are all squares. A triangular pyramid is also called a tetrahedron, Greek for “four faces”.
Right and oblique. A prism is right when its lateral edges stand at right angles to the bases, so one base sits straight above the other; then every lateral face is a rectangle. Otherwise it is oblique: the lateral edges lean, and some lateral faces are parallelograms that are not rectangles. A pyramid is right when its apex is straight above the center of its base, and oblique when it is not.
The height h of a prism is the distance between its bases, measured at right angles to both. The height of a pyramid is the distance from the apex straight down to the level of the base. We write n for the number of sides of the base.
Intuition
Think of a prism as a polygon slid through space without turning, like a cookie cutter pushed straight through dough. Each corner traces a lateral edge, each side sweeps out a lateral face, and where the polygon stops it leaves a copy of itself, the second base. A pyramid joins every point of a base to one point above it, as if the sides were pulled up to a peak. Figure 1 builds both on the same pentagon.
Bases are found by shape, not by position. In everyday speech a base is what a thing stands on. Here the bases are special faces of the solid, wherever they happen to be. A prism lying on its side (Figure 2) rests on a lateral face, and its bases are at the two ends.
Push the top of a right prism sideways while keeping it level, and the lateral edges lean: the prism becomes oblique (Figure 3). Sliding a pyramid’s apex off center does the same (Figure 4). The same ideas built on a circle give a cylinder and a cone, the subject of Cylinders and cones.
Results
For a prism whose bases each have n sides:
For a pyramid whose base has n sides:
The lateral faces of a prism are parallelograms, and rectangles when the prism is right. In a right prism each lateral edge is as long as the height; in an oblique prism each is longer.
Why it is true
The counts. Count in layers, as in Polyhedra. A prism has its two bases and one lateral face for each side of a base, so F = n + 2. Its vertices are the n corners of each base, so V = n + n = 2n. Its edges are n round each base and n lateral edges, one from each corner to its match, so E = 3n.
A pyramid has its base and one triangle for each side of the base, so F = n + 1. Its vertices are the n corners of the base and the apex, so V = n + 1. Its edges are n round the base and n running up to the apex, so E = 2n.
The lateral faces. The second base is the first slid without turning. Every corner makes the same slide, so all the lateral edges are parallel and the same length. A lateral face has two lateral edges as one pair of opposite sides, and a side of a base and its copy as the other pair, also parallel. A four-sided shape whose opposite sides are parallel is a parallelogram. When the slide is straight up, each lateral edge meets the base sides at right angles, so every corner of a lateral face is a right angle and the face is a rectangle.
Height and lateral edges. The height is the shortest way from one base to the other, straight across. A lateral edge of a right prism takes exactly that path, so it equals the height. In an oblique prism it crosses the same gap at a slant, and a slanting path is longer, just as a ladder leaning on a wall is longer than the height it reaches.
Worked examples
A glass prism on the bench
In the science lab, Pavla shines a torch through a glass prism lying on the bench (Figure 2). It rests on a rectangle. Its two ends are triangles, and three rectangles run along its length. Which faces are the bases? Name the solid and say whether it is right or oblique.
SolutionThe bases are the two triangle ends: they are exact copies, they face each other in parallel planes, and each rectangle joins a side of one to the matching side of the other. The rectangle on the bench is a lateral face; how the prism happens to lie does not change its bases. No two rectangles could be the bases instead, since any two of them meet along an edge rather than facing each other.
The bases are triangles, so this is a triangular prism. Every lateral face is a rectangle, so the lateral edges meet the bases at right angles and the prism is right.
Two wedges of cheese
Rutger works at a deli. He stands a block of cheese shaped like a rectangular prism on the counter and cuts straight down along a diagonal of the top. (a) Name each half and find its bases. (b) He also sells wedges from a round wheel of cheese, made by two straight cuts from the center out to the rind. Is such a wedge a prism?
Solution(a) The cut splits the top rectangle into two triangles, and because the knife went straight down, it splits the bottom rectangle along the matching diagonal. So each half has a triangle on top and an exact copy underneath, and the top and bottom of the block are parallel. Its other three faces, two sides of the block and the cut face, are rectangles. Each half is a right triangular prism standing on a base: here the bases are on top and underneath, while on Pavla’s bench they were at the ends.
(b) No. A prism’s whole surface must be flat polygons, and the rind is curved.
Two sandcastle molds
Yaroslav’s tower mold makes a solid with two matching octagons, one at the top and one at the bottom, joined by eight rectangles. His roof mold makes an octagon with eight triangles meeting at a point straight above its center. Name each solid and find F, V and E.
SolutionThe tower is a right octagonal prism: two matching octagons in parallel planes joined by rectangles, with n = 8.
In layers, the edges are 8 round the bottom, 8 round the top and 8 standing upright.
The roof is a right octagonal pyramid, again with n = 8.
On the same base, the prism has 1 more face, 7 more vertices and 8 more edges.
A skeleton from toothpicks
Nevin builds skeleton models: each toothpick is an edge and each mini marshmallow a vertex. He has 36 toothpicks. (a) He wants a prism that uses every toothpick. How many sides does its base have, how many marshmallows does he need, and how many faces does it have? (b) What if he builds a pyramid with all 36? (c) Could a prism use exactly 40 toothpicks?
Solution(a) A prism with an n-sided base has 3n edges, so 3n = 36. Division undoes multiplication by 3, so n = 36 \div 3 = 12: the base is a 12-sided polygon, a dodecagon. He needs V = 2 \times 12 = 24 marshmallows, and the prism has F = 12 + 2 = 14 faces.
(b) A pyramid has 2n edges, so n = 36 \div 2 = 18, and he needs V = 18 + 1 = 19 marshmallows.
(c) No. A prism’s edge count 3n is always a multiple of 3, and 40 \div 3 = 13 remainder 1.
A cube that leans
Kirra builds a cube skeleton from 6 cm toothpicks. The marshmallow joints are soft, so when she pushes the top square to the right it slides over and stays level, and the four upright toothpicks lean. Treat the model as the solid it outlines. (a) Is it still a prism, and is it right or oblique? (b) Is its height still 6 cm?
Solution(a) The top and bottom are still matching squares, and the top stayed level, so they lie in parallel planes. Every top corner moved by the same slide, so the lateral edges are still parallel and the lateral faces are still parallelograms. It is still a square prism, now oblique, because its lateral edges no longer stand at right angles to the bases. The front and back faces are now parallelograms that are not rectangles. The left and right faces only tip over, the way a book tips sideways on a shelf, so they are still rectangles.
(b) No. Each lateral toothpick is still 6 cm long but now crosses the gap at a slant, which is longer than the straight-up distance. The height is now less than 6 cm.
A staircase mold
Yaroslav’s newest mold makes a solid staircase of four steps. Seen from the side, its outline runs along the bottom, up the back, then down four treads and four risers to the front. That outline is the same all the way across, and the two sides are flat. Is it a prism? Find F, V and E, then check F by listing the faces.
SolutionThe two flat sides are exact copies of the outline, facing each other in parallel planes, and every other face is a rectangle joining matching sides. So it is a right prism standing on a lateral face. Its base is the outline, with n = 1 + 1 + 4 + 4 = 10 sides (bottom, back, treads, risers).
Check. The faces are 2 sides, the bottom, the back, 4 treads and 4 risers: 2 + 1 + 1 + 4 + 4 = 12. The base has dents and unequal sides, but the counting used only its number of sides.
Boundaries
The base must be a polygon. Built on a circle, a prism becomes a cylinder and a pyramid a cone. Their surfaces curve, so they are not polyhedra and the counts above do not apply.
A prism’s bases must be exact copies in parallel planes. If the top polygon is smaller than the bottom one, the side faces cannot be parallelograms and the solid is not a prism.
“Right” for a pyramid needs a center. Straight above the center makes sense for a base with a clear center, such as a rectangle or a polygon with all sides and all angles equal.
Sometimes the base is a choice. In a rectangular prism each pair of opposite faces is matching and parallel, so any such pair can be the bases. In a tetrahedron every face is a triangle, so any face can be the base, with the opposite corner as the apex. For most prisms and pyramids, such as a pentagonal prism or a square pyramid, only one choice works.
Where this goes next
Cutting a prism along some edges and laying it flat gives its bases and a band of lateral faces, the subject of Nets of solids; adding the areas of the pieces is Surface area, and how it compares with volume. Filling a box with unit cubes, even when its edges are fractions, is Volume of a box with fraction edge lengths, which grows into Volume of prisms and cylinders: base area times height, for oblique prisms too, as Cavalieri's principle explains. A pyramid holds a third of the prism with the same base and height (Volume of pyramids and cones). Slicing these solids is Cross sections of solids, and the tetrahedron and cube return among the The five regular solids.
Exercises
Names and parts
- Name each solid, and say whether it is right or oblique. (a) Two matching hexagons in parallel planes joined by six rectangles. (b) A seven-sided base and seven triangles meeting at a point straight above the center of the base. (c) Two matching triangles in parallel planes joined by three parallelograms, none of them a rectangle. (d) A solid with six square faces.
- Name each solid in Figure 5, and say whether it is right or oblique.
- Kirra’s pyramid tea bag has four faces, all triangles. Name the solid. Which face is its base?
- A tissue box is a rectangular prism whose length, width and height are all different. How many different pairs of its faces could be called the bases?
- An unsharpened pencil with six flat sides lies on a desk. Name the solid. Which faces are its bases, and what kind of face touches the desk?
- A sandcastle mold has a square bottom, a smaller square top parallel to it, and four trapezoid sides. Is the solid it makes a prism, a pyramid, or neither?
Counting
For Exercises 7 to 10, find F, V and E.
- A prism whose bases each have 7 sides.
- A pyramid whose base has 9 sides.
- A prism whose bases each have 15 sides.
- A pyramid whose base has 16 sides.
- A prism and a pyramid both have 13-sided bases. Which has more edges, and how many more?
Working backward
- A prism has 33 edges. How many sides does its base have? Find F and V.
- A pyramid has 23 faces. How many sides does its base have? Find V and E.
- A prism has 50 vertices. How many sides does its base have? Find F and E.
- Can a pyramid have exactly 15 edges? Can a prism?
- A solid is a prism or a pyramid. It has 18 vertices and 34 edges. Which is it, and how many sides does its base have?
Right and oblique
- A prism’s lateral edges are each 15 cm long and its height is 12 cm. Is it right or oblique?
- A right pentagonal prism has lateral edges 8.5 cm long. What is its height?
- A square pyramid has lateral edges of 7 cm, 7 cm, 9 cm and 9 cm. Can it be a right pyramid? Explain.
Reasoning
- Explain why every prism has an even number of vertices.
- Explain why every pyramid has as many faces as vertices.
- Explain why the number of edges of a prism is always a multiple of 3.
Answers
- (a) A right hexagonal prism. (b) A right heptagonal pyramid (seven-sided base), since the apex is above the center. (c) An oblique triangular prism: lateral faces that are not rectangles mean the lateral edges lean. (d) A cube, a right square prism.
- A: a right hexagonal pyramid. B: a right pentagonal prism lying on a lateral face, its bases at the ends. C: an oblique triangular prism. D: a right triangular pyramid, a tetrahedron.
- A triangular pyramid, or tetrahedron. Any face can be the base: the other three are triangles meeting at the opposite corner, which is the apex.
- 3 pairs: top and bottom, front and back, left and right. Each pair is matching and parallel, joined by four rectangles.
- A right hexagonal prism. Its bases are the two hexagon ends; the face on the desk is a lateral face, a rectangle.
- Neither. The two squares are different sizes, so they are not exact copies and it is not a prism. Its lateral faces do not meet at an apex, so it is not a pyramid; it is like a square pyramid with its top sliced off.
- F = 9, V = 14, E = 21, from F = 7 + 2, V = 2 \times 7, E = 3 \times 7.
- F = 10, V = 10, E = 18, from F = 9 + 1, V = 9 + 1, E = 2 \times 9.
- F = 17, V = 30, E = 45, from F = 15 + 2, V = 2 \times 15, E = 3 \times 15.
- F = 17, V = 17, E = 32, from F = 16 + 1, V = 16 + 1, E = 2 \times 16.
- The prism, by 13: it has 3 \times 13 = 39 edges, the pyramid 2 \times 13 = 26, and 39 - 26 = 13.
- n = 11, F = 13, V = 22. From 3n = 33, n = 33 \div 3 = 11.
- n = 22, V = 23, E = 44. From n + 1 = 23, n = 22.
- n = 25, F = 27, E = 75. From 2n = 50, n = 50 \div 2 = 25.
- A pyramid cannot: its edge count 2n is even, and 15 is odd. A prism can: 3n = 15 gives n = 5, a pentagonal prism.
- A pyramid with a 17-sided base. A prism with 18 vertices would have n = 9 and 27 edges, not 34. A pyramid with 18 vertices has n = 17 and E = 2 \times 17 = 34.
- Oblique. In a right prism each lateral edge equals the height; these are longer, so they lean.
- 8.5 cm. In a right prism each lateral edge runs straight across from base to base, so it equals the height.
- No. In a right square pyramid the apex is straight above the center, and the four corners are the same distance from the center, so the four lateral edges are equal. These are not, so the pyramid is oblique.
- Every vertex of a prism is a corner of one of its two bases, each with n corners, so V = 2n, which is even.
- There is one lateral face per side of the base, plus the base: n + 1 faces. There is one vertex per corner of the base, plus the apex: n + 1 vertices.
- The edges come in three groups of n: round one base, round the other, and the lateral edges joining matching corners. So E = 3n, a multiple of 3.