Cube: Faces, Nets, Volume, Diagonals and Cross-Sections

What is a cube, and how many faces, edges and corners does it have?

A cube is a solid bounded by six identical squares, with 12 edges all of the same length a and 8 corners, and three squares meeting at each corner. It is the regular hexahedron, and one of the five Platonic solids: the only one whose faces are squares. Opposite faces are parallel, every angle between two edges that meet is a right angle, and the length of the edge fixes everything else.

The counts obey Euler’s formula: V-E+F=8-12+6=2, the same for every solid without holes, whether a cube, a pyramid or a soccer ball. Put the cube in coordinates with one corner at the origin and the opposite corner at (1,1,1), and the eight corners are exactly the eight triples of 0s and 1s. An edge joins two corners that differ in one coordinate, so each corner has 3 edges, and (8·3)/(2)=12. That small fact, that a cube is the three-bit binary numbers drawn in space, is used again in the code below (binary numbers explains bits).

How many nets does a cube have, and which arrangements of squares fold into a cube?

A net of the cube is an arrangement of six squares joined edge to edge that folds into a cube with every face covered once; there are exactly 11 different nets. Turned and flipped copies count as the same net. Shapes made of six squares are called hexominoes, and there are 35 of them; only 11 are nets. They come in four families, named by the number of squares in each row: 1-4-1 (a row of four with one square above and one below, in 6 ways), 2-3-1 (3 ways), 2-2-2 (a staircase, 1 way) and 3-3 (1 way).

Why are the other 24 not nets? 12 are ruled out at once: 8 contain a 2×2 block, which cannot lie flat on a cube (going once round a corner brings you back on a different face), and 4 have a row of five or six. The remaining 12 fail by overlap: two squares would land on the same face, leaving one face bare. A straight row shows how: folded round the cube, the squares four apart land on the same face.

A way to test any arrangement is to roll a cube over it: put the cube on the first square, roll it from square to square, and note which face touches each one; it is a net exactly when the six squares meet six different faces. On a net, squares with exactly one square between them in a row are opposite faces. So if you number the squares like a die, opposite faces adding to 7, the pairs fall in place. Build your own below, or load the 11 nets and the three common wrong answers.

How do you find the surface area and the volume of a cube?

The surface area of a cube with edge a is 6a^2, six squares of area a^2, and its volume is a^3, the number of unit cubes in a stack a long, a wide and a high. For a=3 cm: one face is 9 cm², the surface is 6·9=54 cm², and the volume is 3^3=27 cm³, since 3 layers of 3×3 unit cubes make 27. A cube is also a box with all three sides equal, so V=a· a· a is length times width times height.

Scaling. If every edge is multiplied by k, a length is multiplied by k, an area by k^2 and a volume by k^3. Doubling the edge (k=2) multiplies the surface by 4 and the volume by 8; tripling it (k=3) gives 9 and 27. A cube of edge 2 has volume 8 and surface 24, while the unit cube has 1 and 6. This is why a large animal needs thick legs: its weight grows with the cube of its size and the strength of a leg with only the square.

Units. Volume uses cubic units, and one cubic decimetre is exactly one litre, so a cube-shaped water tank with an edge of 40 cm =4 dm holds 4^3=64 litres. A cube of edge 1 m is 1000 dm³: 1000 litres.

Going backwards. Given the volume, take the cube root: a cube of volume 216 cm³ has edge 216^(1/3)=6 cm. Slide the edge below and read every measure, including what doubling does.

How do you find the space diagonal of a cube, and what angles do the diagonals make?

The diagonal of a face of a cube with edge a is a√2 and the space diagonal, joining opposite corners through the inside, is a√3: the Pythagorean theorem used twice. A face is a square, so its diagonal is the hypotenuse of a right triangle with both legs a: f^2=a^2+a^2=2a^2. The space diagonal d is the hypotenuse of a second right triangle whose legs are that face diagonal f and the edge a standing up from its end: d^2=f^2+a^2=3a^2. So d=a√3≈1.732a; for a=5 it is 5√3≈8.66. The longest straight rod that fits inside a cube-shaped room of edge 4 m is 4√3≈6.93 m, along the space diagonal.

A cube has 4 space diagonals, all of the same length, and they all pass through the centre, where each is cut in half. Take the corners of the unit cube: the diagonals have directions (1,1,1), (-1,1,1), (1,-1,1) and (1,1,-1). Two of them meet at an angle with \cosθ=(|(1,1,1)·(-1,1,1)|)/(3)=\frac13, so θ=\arccos\frac13≈70.53^\circ. A space diagonal and the face it starts from make the angle \arctan(1)/(√2)≈35.26^\circ, since the diagonal rises a over a horizontal run a√2. From any corner, the other seven corners are at three distances: 3 at a (along the edges), 3 at a√2 (across the faces), and 1 at a√3 (the opposite corner). Those counts 3,3,1 are the coefficients of (1+x)^3.

What shapes do you get when a plane cuts a cube?

A plane cuts a cube in a triangle, a quadrilateral, a pentagon or a hexagon, never in a polygon with more than six sides, because each side of the cut lies in one face and a cube has only six. The cut can also be a square, a rectangle, a rhombus, a trapezoid or a parallelogram, and the special positions give the nicest shapes:

• a plane parallel to a face cuts a square of area a^2;
• a plane parallel to an edge, such as x+y=t, cuts a rectangle that is a high and t√2 wide (a square when t=(a)/(√2)), and through two opposite edges (x+y=a) it is the rectangle a× a√2 of area a^2√2;
• a plane that cuts off a corner, such as x+y+z=t for t\le a, cuts an equilateral triangle, and the corner it removes is a small pyramid of volume (t^3)/(6) (for t=a a sixth of the cube, with the triangle of side a√2 and area (√3)/(2)a^2);
• the same plane moved to the middle, x+y+z=(3a)/(2), perpendicular to a space diagonal and through the centre, cuts a regular hexagon. Its corners are the midpoints of six edges, its side is (a√2)/(2), and its area is 6·(√3)/(4)·\left((a√2)/(2)\right)^2=(3√3)/(4)a^2≈1.299a^2.

A plane such as x+2y+3z=t shows the whole sequence as it moves: a triangle, then a four-sided cut, then a pentagon, then (through the centre) a parallelogram, and back down the other way. Move the plane in the widget: the shape on the right is the cut laid flat, with its true side lengths. Not every polygon appears: no seven-sided cut exists, and a regular hexagon appears only when the plane passes through the centre perpendicular to a space diagonal.

How large is the sphere inside a cube, and the sphere around it?

The largest sphere inside a cube touches the six faces, has radius \frac a2, and fills \fracπ6≈52.4\% of the cube; the sphere through all eight corners has radius (a√3)/(2) and a volume of (π√3)/(2)a^3≈2.72a^3. The inner sphere has diameter a, the edge, so its volume is \frac43π\left(\frac a2\right)^3=\fracπ6a^3 against the cube’s a^3. The outer sphere has the space diagonal a√3 for a diameter, so its radius is (a√3)/(2) and its volume is \frac43π\left((a√3)/(2)\right)^3=(π√3)/(2)a^3, a little over 2.7 cubes. A third sphere, with radius (a)/(√2), just touches the twelve edges, between the two. So the three radii are in the ratio \frac12:(1)/(√2):(√3)/(2), about 0.5:0.707:0.866 times the edge.

What are cube numbers, and what is the cube of a sum?

A cube number is n^3, the volume of a cube with whole-number edge n: 1,8,27,64,125,…, and the cube of a sum (a+b)^3 breaks a cube of edge a+b into eight pieces. Cut such a cube by planes at distance a from the corners: the pieces are one cube a^3, three slabs a^2b, three rods ab^2 and one small cube b^3, so (a+b)^3=a^3+3a^2b+3ab^2+b^3. For a=2 and b=1: 27=8+12+6+1. The coefficients 1,3,3,1 count the pieces, and they come back as the row of Pascal’s triangle (see expanding brackets).

Two more facts about cube numbers. The sum of the first n cubes is a perfect square, the square of the sum of the first n numbers (Nicomachus’s theorem): 1^3+2^3+…+n^3=\left((n(n+1))/(2)\right)^2, so for n=10 it is 55^2=3025. And to find a cube root that is not a whole number, such as the edge of a cube of volume 30, use Newton’s iteration x_k+1=(2x_k+v/x_k^2)/(3) (see Newton’s method): for 30^(1/3) start at 3 and it settles at 3.1072 within a few steps.

Why can you not double the cube with a compass and a ruler?

To double the volume of a cube you must multiply its edge by 2^(1/3)≈1.259921, and a compass and straightedge cannot construct that length from a unit length; this was proved by Pierre Wantzel in 1837. The volume doubles when k^3=2, so k=2^(1/3). A compass and straightedge can only produce lengths that come from whole numbers by adding, subtracting, multiplying, dividing and taking square roots, and each such step at most doubles the degree of the equation the number satisfies, so the degree is always a power of 2. But 2^(1/3) is a root of x^3-2=0, an equation of degree 3 that cannot be broken into smaller ones, and 3 is not a power of 2. So the length cannot be constructed, however clever the drawing.

The legend says the people of Delos were told by an oracle to double the altar of Apollo, a cube, to stop a plague; the problem is called the Delian problem. Greek geometers solved it with other tools (curves, a marked ruler), and a number can be found as accurately as you wish by arithmetic: Newton’s iteration from 1 gives 1, \frac43=1.3333, (91)/(72)=1.2639, 1.259933, 1.2599210500: after four steps the cube of the answer is 2 to nine digits.

How do you work with a cube in a program?

A program treats a cube as the numbers 0 to 7 read as three bits: the bits are the coordinates of a corner, an edge joins two numbers that differ in one bit, and the distances between corners are counted by the bits that differ. For 0/1 coordinates the number of differing bits is exactly the squared distance, so from a corner there are 3 corners at squared distance 1, 3 at 2 and 1 at 3. Walking along edges through all eight corners is the Gray code i⊕(i≫1), where consecutive numbers differ in one bit: 0,1,3,2,6,7,5,4. The programs below print, for a cube of edge 3, the surface area and volume and the two diagonals, then the counts 8 corners, 12 edges and V-E+F=2, the distance counts, and the walk.

from collections import Counter
from itertools import product
from math import sqrt


def measures(a):
    return {
        'surface': 6 * a * a,
        'volume': a ** 3,
        'face diagonal': round(a * sqrt(2), 3),
        'space diagonal': round(a * sqrt(3), 3),
    }


corners = list(product((0, 1), repeat=3))                 # the eight corners: every triple of 0s and 1s
differ = lambda p, q: sum(x != y for x, y in zip(p, q))   # for 0/1 coordinates this is also the squared distance
edges = [(p, q) for i, p in enumerate(corners) for q in corners[i + 1:] if differ(p, q) == 1]

print(measures(3))                      # {'surface': 54, 'volume': 27, 'face diagonal': 4.243, 'space diagonal': 5.196}
print(len(corners), len(edges), len(corners) - len(edges) + 6)    # 8 12 2
print(sorted(Counter(differ(corners[0], q) for q in corners[1:]).items()))   # [(1, 3), (2, 3), (3, 1)]
print([i ^ (i >> 1) for i in range(8)])  # [0, 1, 3, 2, 6, 7, 5, 4]: a walk along edges through all corners

const measures = (a) => ({ surface: 6 * a * a, volume: a ** 3, faceDiagonal: +(a * Math.sqrt(2)).toFixed(3), spaceDiagonal: +(a * Math.sqrt(3)).toFixed(3) })

const corners = Array.from({ length: 8 }, (_, i) => i)   // the bits of i are x, y and z
const differ = (p, q) => [0, 1, 2].filter((b) => ((p >> b) & 1) !== ((q >> b) & 1)).length
const edges = corners.flatMap((p) => corners.filter((q) => q > p && differ(p, q) === 1).map((q) => [p, q]))

measures(3)                                               // { surface: 54, volume: 27, faceDiagonal: 4.243, spaceDiagonal: 5.196 }
;[corners.length, edges.length, corners.length - edges.length + 6]   // [8, 12, 2]
const at = {}
for (const q of corners.slice(1)) at[differ(0, q)] = (at[differ(0, q)] ?? 0) + 1
at                                                        // { '1': 3, '2': 3, '3': 1 }
corners.map((i) => i ^ (i >> 1))                          // [0, 1, 3, 2, 6, 7, 5, 4]

#include <iostream>
#include <cstdio>
#include <cmath>
using namespace std;

// how many bits differ: for corners numbered 0 to 7 this is the squared distance
int differ(int p, int q) {
  int d = 0;
  for (int bit = 0; bit < 3; bit++) {
    if (((p >> bit) & 1) != ((q >> bit) & 1)) d++;
  }
  return d;
}

int main() {
  double a = 3;
  printf("surface %.0f; volume %.0f\n", 6 * a * a, a * a * a);
  printf("face diagonal %.3f; space diagonal %.3f\n", a * sqrt(2.0), a * sqrt(3.0));

  int edges = 0;
  for (int p = 0; p < 8; p++) {
    for (int q = p + 1; q < 8; q++) {
      if (differ(p, q) == 1) edges++;
    }
  }
  printf("%d %d %d\n", 8, edges, 8 - edges + 6);

  int at1 = 0;
  int at2 = 0;
  int at3 = 0;
  for (int q = 1; q < 8; q++) {
    int d = differ(0, q);
    if (d == 1) at1++;
    if (d == 2) at2++;
    if (d == 3) at3++;
  }
  printf("%d %d %d\n", at1, at2, at3);

  for (int i = 0; i < 8; i++) printf("%d ", i ^ (i >> 1));
  printf("\n");
  return 0;
}

/* output:
surface 54; volume 27
face diagonal 4.243; space diagonal 5.196
8 12 2
3 3 1
0 1 3 2 6 7 5 4
*/

How many different dice are there, and how many positions has a Rubik’s cube?

There are 30 different dice if the faces carry 1 to 6 once each and turning a die does not make a new one, and only 2 of them have opposite faces adding to 7, which are mirror images of each other. Without turning there are 6!=720 ways to write the numbers on six faces, and a cube can be turned in 24 ways (the 6 faces that can lie on the table times 4 quarter turns), so (720)/(24)=30. Fix the pairs of opposite faces as (1,6), (2,5), (3,4) and only the left or right handed arrangement remains: place 1 on top and 2 in front, and 3 goes either on the right or on the left. Dice for games use one of the two, the other being their reflection. Dice are very old: cubes marked this way are about 5000 years old.

A 3×3×3 puzzle cube has 8 corner pieces, 12 edge pieces and 6 centres. The corners can be arranged in 8! ways and each can be twisted in 3 ways (the last twist is forced), the edges in 12! ways and each flipped in 2 ways (the last flip is forced), and half of these combinations cannot be reached by turning the faces, giving (8!·3^7·12!·2^11)/(2)=43 252 003 274 489 856 000≈4.3×10^19 positions. Of the 27 little cubes of a 3×3×3 block, 26 show a face and the middle one is hidden, which is also 3^3-1^3.

Where does the cube come from in mathematics?

The cube is among the oldest shapes in mathematics: it is one of the five regular solids the Greeks described, and a problem about it stood open for more than two thousand years.

• About 3000 BCE. Dice shaped like cubes, with the faces marked by dots, are found in the ancient Near East; the oldest known are about 5000 years old.
• About 360 BCE. Plato’s Timaeus associates the five regular solids with the elements, and gives the cube to earth, the most stable.
• About 300 BCE. Euclid’s Elements ends with Book XIII, which constructs the five regular solids; Proposition 15 constructs the cube and encloses it in a sphere.
• About 430 BCE onward. The Delian problem, doubling a cube, is posed. Hippocrates of Chios reduces it to finding two mean proportionals, and Archytas finds a construction in three dimensions.
• 1750. Euler states the formula V-E+F=2 for solids such as the cube, in a letter to Goldbach.
• 1837. Pierre Wantzel proves that doubling the cube, and trisecting an angle, cannot be done with a compass and straightedge.
• 1974. Ernő Rubik, a Hungarian architect, invents the puzzle cube; it is sold around the world from 1980.

What are the common mistakes with the cube?

The most common mistakes with the cube are the nine below, each with the correct statement.

Practice: test your understanding

These questions mix everything above. A wrong answer costs nothing here: read the hint and try again.

Summary: the cube at a glance

• Parts: 6 faces, 12 edges, 8 corners; 8-12+6=2. The corners are the triples of 0s and 1s.
• Area and volume: surface 6a^2, volume a^3; scaling by k gives k^2 and k^3; 1 dm³ is 1 litre.
• Diagonals: face a√2, space a√3; two space diagonals meet at \arccos\frac13≈70.53^\circ.
• Nets: 11 of the 35 hexominoes fold; check by rolling a cube; opposite faces are one square apart in a row.
• Cuts: triangle, quadrilateral, pentagon, hexagon, never more than six sides; the central cut perpendicular to a space diagonal is a regular hexagon of area (3√3)/(4)a^2.
• Spheres: inside, radius \frac a2, fills \fracπ6 of the cube; around, radius (a√3)/(2).
• Cube numbers: (a+b)^3=a^3+3a^2b+3ab^2+b^3; 1^3+…+n^3=\left((n(n+1))/(2)\right)^2; doubling needs 2^(1/3), not constructible.

Glossary: key terms

Cube

A solid with six identical square faces, twelve equal edges and eight corners, three faces meeting at each corner.

Face, edge, vertex

A flat square of the surface, where two faces meet, and where three edges meet.

Net

A flat arrangement of squares joined edge to edge that folds into the cube with no overlap; the cube has 11.

Surface area

The total area of the faces; for a cube with edge a it is 6 a squared.

Volume

The space inside a solid, measured in cubic units; for a cube with edge a it is a cubed.

Space diagonal

The segment joining two opposite corners through the inside of the cube; its length is a times the square root of 3.

Face diagonal

The diagonal of one square face; its length is a times the square root of 2.

Cross-section

The shape in which a plane cuts a solid; for a cube it has between three and six sides.

Cube root

The number whose cube is a given number; the edge of a cube whose volume is given.

Euler’s formula

V minus E plus F equals 2 for a solid without holes, with V vertices, E edges and F faces.

Frequently asked questions

References

1. Peter R. Cromwell (1997). Polyhedra. Cambridge University Press.
2. H. S. M. Coxeter (1973). Regular Polytopes (3rd ed.). Dover.
3. Thomas L. Heath (translator) (1956). The Thirteen Books of Euclid’s Elements (Book XIII). Dover.
4. Pierre Laurent Wantzel (1837). Recherches sur les moyens de reconnaître si un problème de Géométrie peut se résoudre avec la règle et le compas. Journal de Mathématiques Pures et Appliquées.
5. David Joyner (2008). Adventures in Group Theory: Rubik’s Cube, Merlin’s Machine, and Other Mathematical Toys (2nd ed.). Johns Hopkins University Press.
6. Carl B. Boyer, Uta C. Merzbach (2011). A History of Mathematics (3rd ed.). Wiley.
7. Martin Aigner, Günter M. Ziegler (2018). Proofs from THE BOOK (6th ed.). Springer.

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