Cubes and Cube Roots

Cubes and Patterns

What is a cube? What is its volume? How would you define its properties? Did you know that the cube has all three-dimensional parameters of measurements equal? Let’s learn the concepts of a cube along with its properties and patterns.

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What is a Cube?

The cube is a three-dimensional structure which is formed when six identical squares bind to each other in an enclosed form.

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Volume of a Cube

The volume of a cube can be represented by length (l) × breadth (b) × height (h), and since l = b = h in the cube, its sides can be represented as l = b = h = a. Therefore, the volume of the cube = a3, where a is the measurement of each side of the cube. Hence, the volume of the cube of side 1 cm will be equal to 1 cm × 1 cm × 1 cm = 1 cm3.

However, if we require constructing a larger cube from a smaller cube (a = 1 cm), we need to join a number of those smaller cubes. Examples of things that have the shape representation those of a cube are – Dice, Rubik’s Cube, etc. A cube generally has 6 faces, 12 edges, and 8 vertices. The pattern of the cube is also a square parallelepiped or the shape of an equilateral cuboid.

cube

In terms of numbers, Cube numbers or simply put, Cubes, are those special numbers that are obtained by multiplying any given number by itself, three times.

  • Cubes of positive numbers are always positive. For example: Cube of +4 is = (+4) x (+4) x (+4) = +64
  • Cubes of negative numbers are always negative. For example: Cube of -4 is = (-4) x (-4) x (-4) = -64

Numerical value obtained after cubing any given number is called a Perfect Cube. Cubes of some natural numbers are given as per following: –

13 = 1
23 = 8
33 = 27
43 = 64
53 = 125
63 = 216
73 = 343
83 = 512
93 = 729
103 = 1000

While cubing, we observe that perfect cubes are numbers like 1, 8, 27, etc. but numbers falling between these are NOT perfect cubes.

Properties of Perfect Cubes

The cubes of numbers with counting digits, i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 as the unit’s digits end with 0, 1, 8, 7, 4, 5, 6, 3, 2, and 9 respectively. This might be complex and hard to remember but only knowing the cubes of numbers mentioned above, determination of the unit’s digit of any large number’s cube becomes quite easy and this is also very helpful in many calculations.

  • Cubes of even numbers are ALWAYS even.
  • Cubes of odd numbers are ALWAYS odd.

Patterns Related to Cubes

  • A perfect cube is “a sum of consecutive odd numbers”. The consecutive odd numbers, whose sum makes up a perfect cube, themselves appear in the order of odd numbers of the number line. For example: –

13 = 1 = 1
23 = 8 = 3 + 5
33 = 27 = 7 + 9 + 11
43 = 64 = 13 + 15 + 17 + 19
53 = 125 = 21 +33 + 35 + 37 + 39 + 41
And so on.

  • Another interesting pattern found out, is that the count of consecutive odd numbers, whose sum makes up a perfect cube, is equal to the number whose cube is under consideration. For example: –

13 = 1 = 1 (one odd number)
23 = 8 = 3 + 5 (two odd numbers)
33 = 27 = 7 + 9 + 11 (three odd numbers)
43 = 64 = 13 + 15 + 17 + 19 (four odd numbers)
53 = 125 = 21 +33 + 35 + 37 + 39 + 41 (five odd numbers)
And so on.

Now, how to determine if a number is a perfect cube or not? Well, the Prime Factorization method is used for this purpose. We prime factorize the given number and obtain all the possible factors, refer the examples in the section below.

Solved Examples For You

Q. Let the numbers be – 15625, 8000 and 243. Verify if the given numbers are perfect cubes or not.

Ans: Prime factorization of 15625 = 5x5x5x5x5x5 (product of 6 times 5). Now, we make possible groups of 3 like digits; three 5’s make a group and three 5’s make another group.
15625 = (5x5x5) x (5x5x5) = 53 x 53 = (5×5)3 = (25)3
Thus, 15625 is a PERFECT CUBE.

Prime factorization of 8000 = 2x2x2x2x2x2x5x5x5 (product of 6 times 2 and 3 times 5). Now, we make possible groups of 3 like digits; three 2’s make a group, three 2’s make another group and three 5’s make another group.
15625 = (2x2x2) x (2x2x2) x (5x5x5) = 23 x 23 x 53 = (2x2x5)3 = (20)3
Thus, 8000 is also a PERFECT CUBE.

Prime factorization of 243 = 3x3x3x3x3 (product of 5 times 3). Now, we make possible groups of 3 like digits; three 3’s make a group, two 3’s make another group.
243 = (3x3x3) x (3×3) = 33 x 32
Thus, 243 is NOT a PERFECT CUBE as it cannot be written as a cube of a single number.

Ques. How many squares are in a cube?

Ans. We know that every square has four vertices, four edges, and a square face as well. We can form a model of a cube and we can count its eight vertices, twelve edges, and six squares.

Ques. What is the shape of a cube?

Ans. A cube comprises of 6 equal and square-shaped sides. Cubes have 8 vertices and twelve edges, all these are of the same length. The angles in each cube are all right angles (90 degrees). Substances that are cube in shape contains building blocks and dice.

Ques. What are the properties of a cube?

Ans. A cube is a solid that contains 6 square faces all of the same size that meet each other at an angle of 90 degrees. All the edges are of the same length.

Ques. What will be the area of each face of the cube?

Ans. Since all the sides of a cube are equal, the surface area of the cube is six times to the area of each face. For the dimensions, one face has an area of 12 multiplied by 12 or 144 in2. Therefore, the total surface area will be 6 multiplied by 144 that is equal to 864 in2.

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