Cubes 1 To 100: Complete Table, Patterns And Uses

What Is The Cube Of A Number?

The cube of a number is that number multiplied by itself three times, written with the exponent 3. So $n^3 = n \times n \times n$, and $7^3 = 7 \times 7 \times 7 = 343$.

The name comes from geometry: a cube with edge $n$ units has volume $n^3$ cubic units. A number that is the cube of a whole number is a cube number, which is why 343 counts and 344 does not.

The exponents behind the notation carry straight on into algebra, where $x^3$ follows exactly the same rules as $7^3$.

Where Do Cubes 1 To 100 Show Up, And Why Do They Matter?

Cubes show up wherever something grows in three directions at once, so any volume calculation is a cube calculation. A cube of edge 100 cm (one metre) holds $100^3 = 1{,}000{,}000$ cubic centimetres, which is why $1,\text{m}^3 = 1{,}000{,}000,\text{cm}^3$.

They also matter wherever one quantity rises with the cube of another. The power a wind turbine can draw from the wind is proportional to the cube of the wind speed, so doubling the wind speed gives $2^3 = 8$ times the power.

The same scaling explains a quieter fact about models and animals. Make every length 10 times bigger and the volume, and so the weight, grows $10^3 = 1{,}000$ times, while the surface grows only $10^2 = 100$ times.

How Do You Read The Cubes 1 To 100 Table?

Find $n$ in a left-hand column and read the number to its right. For example, $47$ sits in the 26 to 50 block, and the entry beside it is $103{,}823$, so $47^3 = 103{,}823$.

You can also read the table backwards to find a cube root. Spot $157{,}464$ in the 51 to 75 block, look left, and you get $54$, so $\sqrt[3]{157464} = 54$.

Three landmarks make the table easy to scan:

• The thousands step: every multiple of 10 has a cube ending in 000, from $10^3 = 1{,}000$ to $90^3 = 729{,}000$.

• The six-digit line: $46^3 = 97{,}336$ is the last five-digit cube, and $47^3 = 103{,}823$ is the first six-digit cube.

• The finish: $99^3 = 970{,}299$, and $100^3 = 1{,}000{,}000$ is the only seven-digit cube in the table.

Cubes Of 1 To 100

The table runs in four blocks of 25. Read each block down the columns: the left number in a pair is $n$ and the right number is $n^3$.

Cubes Of 1 To 25

Cubes Of 26 To 50

Cubes Of 51 To 75

Cubes Of 76 To 100

How Many Perfect Cubes Are There From 1 To 100?

There are four perfect cubes from 1 to 100: $1 = 1^3$, $8 = 2^3$, $27 = 3^3$ and $64 = 4^3$. The next one, $5^3 = 125$, is already past 100.

A perfect cube is a whole number whose cube root is also a whole number. Only two numbers from 1 to 100 are both perfect cubes and perfect squares: $1$ and $64 = 8^2 = 4^3$.

The count grows slowly as the range widens, because cubes spread out fast:

Which Cubes From 1 To 100 Are Even And Which Are Odd?

An even number always has an even cube, and an odd number always has an odd cube. So the table holds 50 even cubes (from $2^3 = 8$ to $100^3 = 1{,}000{,}000$) and 50 odd cubes (from $1^3 = 1$ to $99^3 = 970{,}299$).

The reason is short. An odd number times an odd number is odd, so three odd factors stay odd; one factor of 2 in $n$ puts at least $2^3 = 8$ into $n^3$, which makes every even cube a multiple of 8.

What Patterns Can You See In Cubes 1 To 100?

The table is full of patterns, and each one doubles as a check on a cube you have just worked out. Here are the five that do the most work.

The Units Digit Pattern

The units digit of $n^3$ depends only on the units digit of $n$, and every digit 0 to 9 maps to a different units digit:

Digits 0, 1, 4, 5, 6 and 9 repeat themselves, while 2 and 8 swap and 3 and 7 swap. That one-to-one match is why the units digit of a perfect cube tells you the units digit of its cube root.

Cubes As Sums Of Consecutive Odd Numbers

Each cube is the sum of $n$ consecutive odd numbers, and the odd numbers simply carry on from one cube to the next:

$$1^3 = 1$$

$$2^3 = 3 + 5 = 8$$

$$3^3 = 7 + 9 + 11 = 27$$

$$4^3 = 13 + 15 + 17 + 19 = 64$$

$$5^3 = 21 + 23 + 25 + 27 + 29 = 125$$

The block for $n^3$ starts at the odd number $n^2 - n + 1$. This builds on the sum of odd numbers pattern for squares, and school textbooks often use it as the first cube pattern.

Gaps Between Neighbouring Cubes

The gap between $n^3$ and $(n+1)^3$ is $3n^2 + 3n + 1$:

$$(n+1)^3 - n^3 = 3n^2 + 3n + 1$$

The first gaps are 7, 19, 37, 61, 91, and they keep widening. That is why perfect cubes thin out so quickly as numbers grow.

Trailing Zeros Come In Threes

A cube of a multiple of 10 ends in exactly three times as many zeros as the number. So $30^3 = 27{,}000$ (one zero becomes three), and $100^3 = 1{,}000{,}000$ (two zeros become six).

Digital Roots Of 1, 8 Or 9

Add the digits of any cube, and keep adding until one digit is left. The answer is always 1, 8 or 9: for example $47^3 = 103{,}823$ gives $1+0+3+8+2+3 = 17$, then $1 + 7 = 8$.

A result of 2, 3, 4, 5, 6 or 7 means the number cannot be a cube. It is a fast check on a long multiplication, though it is a filter and not a proof, since many non-cubes also have digital root 1, 8 or 9.

What Is The Sum Of Cubes From 1 To 100?

The sum of the cubes from 1 to 100 is $25{,}502{,}500$. It comes from the formula for the sum of cubes of n natural numbers:

$$1^3 + 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$

With $n = 100$:

$$\frac{100 \times 101}{2} = 5050$$

$$5050^2 = 25{,}502{,}500$$

The formula says something surprising: the sum of the first $n$ cubes equals the square of the sum of the first $n$ whole numbers. Check it small, with $1^3 + 2^3 + 3^3 = 36$ and $(1 + 2 + 3)^2 = 6^2 = 36$.

How Do You Calculate The Cube Of Any Number From 1 To 100?

Multiplying three times always works, but two-digit numbers go faster with the binomial cube. The table above is the answer key for both.

Method 1: Multiply Three Times

Square the number first, then multiply by the number again:

$$37^2 = 1369$$

$$37^3 = 1369 \times 37 = 50{,}653$$

Squaring first keeps every step a two-number multiplication, which is where the fewest errors creep in.

Method 2: Split The Number And Use The Binomial Cube

Write a two-digit number as tens plus units, $10a + b$, and use the cube of a binomial:

$$(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$$

For $23 = 20 + 3$:

$$20^3 = 8000$$

$$3 \times 20^2 \times 3 = 3600$$

$$3 \times 20 \times 3^2 = 1080$$

$$3^3 = 27$$

$$23^3 = 8000 + 3600 + 1080 + 27 = 12{,}167$$

Method 3: The Four-Column Trick For Two-Digit Numbers

This is the "cube numbers 1 to 100 trick" that circulates for competitive exams, and it is Method 2 laid out in columns. For a number with tens digit $a$ and units digit $b$, write four columns: $a^3$, then $3a^2b$, then $3ab^2$, then $b^3$.

For $47$ ($a = 4$, $b = 7$) the columns are $64$, $336$, $588$, $343$. Now work from the right, keeping one digit and carrying the rest:

1. $343$: write 3, carry 34.

2. $588 + 34 = 622$: write 2, carry 62.

3. $336 + 62 = 398$: write 8, carry 39.

4. $64 + 39 = 103$: write 103.

Read the digits: $47^3 = 103{,}823$, which matches the table.

Method 4: Numbers Close To 100

For $100 - k$, expand $(100 - k)^3$ and only small numbers are left to handle:

$$(100 - k)^3 = 1{,}000{,}000 - 30{,}000k + 300k^2 - k^3$$

For $98$, $k = 2$:

$$1{,}000{,}000 - 60{,}000 + 1200 - 8 = 941{,}192$$

How Do You Estimate A Cube In Your Head?

Round to the nearest multiple of 10 and cube that, then adjust in the right direction. For $47^3$, $50^3 = 125{,}000$ is too big, and since $47$ is 6% smaller than 50, the cube is about 18% smaller (three times 6%), giving roughly $102{,}500$ against the true $103{,}823$.

That 3-times rule for small percentage changes is a handy mental math habit for any cube. It drifts for bigger changes, so treat it as a first guess and check with the table.

How Are Cubes Different From Cube Roots?

Cubing and taking the cube root undo each other: $4^3 = 64$ and $\sqrt[3]{64} = 4$. This table goes from $n$ to $n^3$, and reading it right to left goes from a perfect cube back to its root.

A cube root of a number that is not a perfect cube is irrational, so it never appears as a whole number in this table. For example, 100 sits between $4^3 = 64$ and $5^3 = 125$, so $\sqrt[3]{100} \approx 4.6416$.

Cubes are also easy to confuse with squares. The square of 6 is 36 and the cube of 6 is 216; the squares from 1 to 50 table grows far more slowly than this one.

Examples Of Cubes 1 To 100

Example 1

A storage box is a cube with edge 75 cm. What is its volume?

$$V = 75^3$$

$$75^2 = 5625$$

$$5625 \times 75 = 421{,}875$$

Final answer: $421{,}875,\text{cm}^3$, which is about 0.42 cubic metres.

Example 2

Find $30^3$.

A quick first attempt: $3^3 = 27$, then attach the zeros from 30 to get $2{,}700$.

Check the size. $30^3$ is $30 \times 30 \times 30$, and already $30 \times 30 = 900$, so the answer must be about 30 times 900, far more than 2,700.

The zero in 30 is cubed along with the 3, so it appears three times:

$$30^3 = 3^3 \times 10^3 = 27 \times 1000 = 27{,}000$$

Final answer: $27{,}000$

Example 3

Is 9,261 a perfect cube? If so, find its cube root.

The units digit is 1, so a cube root would end in 1. Also $20^3 = 8000$ and $30^3 = 27{,}000$, so the root lies between 20 and 30.

The only candidate is 21:

$$21^2 = 441$$

$$441 \times 21 = 9261$$

Final answer: Yes, $9261 = 21^3$, so its cube root is 21.

Example 4

A cube-shaped tank holds 157,464 cubic centimetres. How long is each edge?

The units digit 4 means the edge ends in 4. Since $50^3 = 125{,}000$ and $60^3 = 216{,}000$, the edge is between 50 and 60, so it must be 54.

$$54^3 = 2916 \times 54 = 157{,}464$$

Final answer: 54 cm

Example 5

Find $1^3 + 2^3 + 3^3 + \cdots + 20^3$.

$$\frac{20 \times 21}{2} = 210$$

$$210^2 = 44{,}100$$

Final answer: $44{,}100$

Example 6

Which whole numbers from 1 to 100 have cubes between 100,000 and 200,000?

From the table, $46^3 = 97{,}336$ is just below 100,000 and $47^3 = 103{,}823$ is the first cube above it.

At the top, $58^3 = 195{,}112$ is still below 200,000, while $59^3 = 205{,}379$ is over.

Final answer: the twelve numbers 47 to 58.

Examples 3 and 4 are where the patterns earn their keep. Using the units digit to fix the last digit and the cubes of 10s to fix the first one turns a long division search into two quick look-ups, and it is the habit that holds up when there is no table in front of you.

Where Do Students Go Wrong With Cubes 1 To 100?

Mistake 1: Reading $n^3$ As $3n$

Where it slips in: Early practice, when exponents are new.

Don't do this: Writing $5^3 = 15$ because the small 3 "means times three".

The correct way: $5^3 = 5 \times 5 \times 5 = 125$. The exponent counts how many 5s are multiplied, not what 5 is multiplied by; this is the first instinct for many students meeting powers, and saying "5 times 5 times 5" out loud fixes it faster than any rule.

Mistake 2: Dropping Zeros From Multiples Of 10

Where it slips in: Cubing 20, 30, 40 and so on mentally.

Don't do this: Writing $40^3 = 6{,}400$ by cubing the 4 and keeping two zeros, which is really $80^2$.

The correct way: Each zero is cubed along with the digit, so it appears three times: $40^3 = 64 \times 1000 = 64{,}000$. The rusher who has just practised squares, where one zero becomes two, tends to carry that count over to cubes.

Mistake 3: Cubing A Sum Term By Term

Where it slips in: Using Method 2 on a two-digit number.

Don't do this: Writing $23^3 = 20^3 + 3^3 = 8027$.

The correct way: $(20 + 3)^3$ has four terms, $20^3 + 3 \times 20^2 \times 3 + 3 \times 20 \times 3^2 + 3^3 = 12{,}167$. The memorizer who learned $a^3 + b^3$ as a formula name often expands $(a + b)^3$ into it, so check the size: 8,027 is far below $23^3$, which must exceed $20^3 = 8000$ by a lot.

Where Can You Get Help With Cubes And Algebra?

The table is quick to look up, but the patterns behind it, binomial cubes and the sum formula, are where cubes connect to algebra. To build those links with a teacher, look at an algebra tutor, a middle school math tutor, mental maths for kids or math classes online.

Conclusion

• Cubes 1 to 100 are the values of $n^3$ for $n = 1$ to $100$, running from 1 to 1,000,000.

• Only four perfect cubes lie between 1 and 100: 1, 8, 27 and 64.

• The units digit of a cube depends only on the units digit of the number, which makes it the fastest check on any cube.

• The sum of cubes 1 to 100 is $5050^2 = 25{,}502{,}500$, the square of the sum of 1 to 100.

• Two-digit cubes come fastest from $(10a + b)^3$, and the most common errors are reading $n^3$ as $3n$ and dropping zeros.

Practice These Before Moving On

1. Find $12^3$ and $25^3$ without the table.

2. Use the four-column trick to find $36^3$.

3. Is 21,952 a perfect cube? If so, find its cube root.

4. Find $1^3 + 2^3 + \cdots + 50^3$.

5. Use the near-100 expansion to find $99^3$.

Answers:

1. $12^3 = 1728$ and $25^3 = 15{,}625$

2. Columns $27$, $162$, $324$, $216$ give $46{,}656$

3. Yes; units digit 2 points to 8, and between $20^3$ and $30^3$ that means 28, and $28^3 = 21{,}952$

4. $\frac{50 \times 51}{2} = 1275$ and $1275^2 = 1{,}625{,}625$

5. $1{,}000{,}000 - 30{,}000 + 300 - 1 = 970{,}299$

If question 2 trips you up, go back to Method 3 and redo the carries one column at a time.

Want a live trainer to walk through more problems with cubes? Book a free demo class

Read More

• Cube numbers

• Perfect cube

• Cube root 1 to 100

• Cube root of 343

• Cube root of 1331