The Cardboard Computer

How to use the Cardboard Computer

There are two versions of the Cardboard Computer available: the basic version has just the D and C scales, which allow multiplication and division. This version is suitable for beginners and for pre-secondary-school students; the advanced version adds the CI, A, and K scales for three-step multiplication, square roots, and cube roots. This version is suitable for secondary or post-secondary math users.

Quick start 🔗

The Cardboard Computer is set up so that the numbers all around the outer scales (D and C) will be in the same proportion to each other no matter how you turn the wheels. For example, if 2 on the D scale is above 1 on the C scale, then 4 will be above 2 and so on:

D scale 2 3 4 5 5 7 8 9
C scale 1 1.5 2 2.5 3 3.5 4 4.5

To divide, rotate the wheels to place the divisor above the dividend (just as if you were writing a fraction), then read the quotient on the D scale above 1 (the "unit pointer") on the C scale. With the wheels set up as in the above table, 4 ÷ 2, 6 ÷ 3, and 8 ÷ 4 all equal 2.

To multiply, place the first factor on the D scale above one, then read the product on the D scale above the second factor on the C scale. So, using the same diagram (where you've rotated the wheels to place 2 above 1), 2 × 2 = 4, 2 × 3 = 6, and 2 × 4 = 8.

For bigger or smaller numbers, move the decimals around as needed, and things still work (e.g. 80 ÷ 40 is still 2, while 80 ÷ 4 is 20). You can picture either scale like this (extending indefinitely in both directions).

0.01 0.011 0.012 0.02 0.025 0.03
0.1 0.11 0.12 0.2 0.25 0.3
1 1.1 1.2 2 2.5 3
10 11 12 20 25 30
100 110 120 200 250 300

If you've never used a slide rule before, it's a good idea to play around with multiplication and division for a while until you're comfortable with it. You can also watch some animated demos for solving basic problems.

1. Overview of the scales 🔗

Scale Uses Level
D Multiplication and division, and ratios and conversions (with the C scale). Basic
C Multiplication and division, and ratios and conversions (with the D scale). Basic
CI Combined calculations. Advanced
A Squares and square roots (with the C scale). Advanced
K Cubes and cube roots (with the C scale). Advanced
Illustration of a circular slide rule with D, C, CI, A, and K scales.

2. Multiplication and division 🔗

The D and C scales are spaced so that at any point, the number on the D scale and the number immediately below on the C scale are in the same proportion: for example, if 4 appears above 2, then 6 will appear above 3 and 8 will appear above 4. This relationship allows for basic multiplication and division by putting one of the numbers above 1 (the unit pointer, with a black background).

2.1. Multiplying two numbers 🔗

Apply the order-of-magnitude rules to the following if/as needed.

  1. Find the first number (factor) on the D scale and rotate so that it appears above the black unit pointer (1) on the C scale.
  2. Preserving the position of the wheels, find the second number (factor) on the C scale
  3. Read the result (product) directly above on the D scales.

Example: 41 × 12 = ?

  • Applying the order-of-magnitude rules, divide 41 by 10 once to get 4.1, and divide 12 by 10 once to get 1.2, so that both fall in the range 1–10.
  • Rotate the wheels until 4.1 on the outer D scale is directly above the unit pointer (1) on the inner C scale.
  • Without turning the wheels, find 1.2 on the C scale.
  • Read the number 4.92 on the D scale directly above 1.2 on the C scale. Use the cursor to help, if you added one.
  • Again, applying the order-of-magnitude rules, multiply the result 4.92 by 10 twice to get the answer 492.

2.2. Dividing two numbers 🔗

Apply the order-of-magnitude rules to the following if/as needed.

  1. Find the first number (dividend) on the D scale.
  2. Rotate the wheels until the second number (divisor) appears on the C scale directly underneath the first number.
  3. Preserving the position of the wheels, read the result (quotient) on the D scale above the unit pointer (1) on the C scale.

Example: 15 ÷ 4 = ?

  • Find 1.5 (for 15) on the outer D scale.
  • Rotate the wheels until 4 on the inner C scale is directly beneath 1.5.
  • Without turning the wheels, read the answer 3.75 on the D scale directly above the unit pointer (1).

2.3. Multiplication followed by division 🔗

Apply the order-of-magnitude rules to the following as needed.

  • Find the first factor on the D scale.
  • Turn the wheel so that the first factor is directly above the divisor on the C scale.
  • Without moving the wheels, find the second factor on the C scale.
  • Read the answer directly above that on the C scale.

Example: 5π ÷ 9.5 = ?

  • Find the first factor, 5 on the D scale.
  • Turn the wheels so that 5 is directly above the divisor, 9.5, on the C scale.
  • Without moving the wheels, find the second factor, π (a special marking just past 3) on the C scale.
  • Read the approximate answer 1.66 directly above on the D scale.

3. Ratios and conversions 🔗

Since ratios are consistent all around the C and D scales, you can use them to convert numbers (e.g. distance, currency), adjust things by ratios (e.g. tripling a recipe), and so on. The nice thing is that you need to set the wheel only once, and then you can keep reading different numbers from it.

  1. Set the first number of the ratio on the outside (D) scale.
  2. Set the second number of the ratio on the inside (C) scale.
  3. Read a number anywhere on either scale, and find its equivalent directly above or below on the other scale.

3.1. Simple conversions 🔗

The simplest conversions are ratios one of the sides is 1 (e.g. 1 mile == 1.6 kilometers, or €1.00 == £1.17). In that case, place the other value over the unit pointer (1) once, and the Cardboard computer becomes a static calculator where you can read results all around the scale without moving the wheel.

Example 1: convert 8 miles to kilometres

  • Place 1.6 on the D scale above the unit pointer (1) on the C scale.
  • Find 8 (miles) on the C scale.
  • Read the approximate value 12.8 km (1.28) above on the D scale.

Example 2: convert 25 Euros (€) to Pounds Sterling (£)

  • (Assume that the conversion rate is £1.00 to €1.17.)
  • Place 1.17 on the D scale directly above the unit pointer (1) on the C scale.
  • Find 25 (2.5) on the D scale.
  • Read the approximate result €21.40 (2.14) directly below on the C scale.

Example 3: triple a recipe

  • Place 3 on the D scale above the unit pointer (1) on the C scale.
  • For each ingredient, find the original value on the C scale and read the new value on the D scale.
  • E.g., if the original recipe had 4.5 cups of water, the tripled recipe will have 13.5 (1.35) cups of water.

3.2. Other ratios 🔗

Sometimes ratios do not involve 1. Those work exactly the same way, except that you don't use the unit pointer. For example, old cathode-ray TV screens usually had a picture ratio of 4:3, while newer high-resolution flat-screen TVs have a picture ratio of 16:9. Odds in betting are also typically given with two arbitrary integers.

Example 1: calculate the height of a screen with 16:9 aspect ratio, if the screen is 60cm wide.

  • Place 16 on the D scale above 9 on the C scale.
  • Find 60 (6.0) on the D scale.
  • Read the approximate height, 33.8 cm, directly below on the C scale.

Example 2: calculate how much a friendly $15 wager will pay at 8:3 odds

  • Place 8 on the D scale above 3 on the C scale.
  • Find $15 (1.5) on the D scale.
  • Read the approximate extra return (on top of your original bet), $5.63, directly below on the C scale.
  • (Repeat for any other values you need to check, without moving the wheels.)

Many artists and architects consider the Golden Ratio (approximately 5:3) to be visually appealing. It is easy to estimate using the Cardboard computer.

Example 3: if the main section of a web page is 800 px wide, how wide should the sidebar be to maintain the Golden Ratio?

  • Place 5 on the D scale above 3 on the C scale.
  • Find 800 (8.0) on the D scale.
  • Read the result, 480 px, directly below on the C scale.

3.3. Temperature conversions 🔗

1.8° Fahrenheit is equal to 1° Celsius, so the conversion ratio on the Cardboard Computer is 1.8:1. However, because the scales use different values for freezing, there is one extra step: you have to subtract 32 before converting from Fahrenheit to Celsius, or add 32 after converting from Celsius to Fahrenheit.

Example 1: convert 15°c to Fahrenheit

  • Set the ratio 1.8 on the D scale over the unit pointer (1) on the C scale.
  • Find 15°c (1.5) on the C wheel.
  • Read 27 (2.7) on the D scale directly above 15 (1.5) on the C scale.
  • Add 32 to 27 to get 59°f.

Example 2: convert 20°f to Celsius

  • Set the ratio 1.8 on the D scale over the unit pointer (1) on the C scale.
  • Subtract 32 from 20 to get -12.
  • Find 12 (1.2) on the D scale.
  • Read the answer -6.7°c (6.7) on the C wheel directly below -12 (1.2) on the D wheel.

4. Square and cube roots (advanced only) 🔗

While the C and D scales each represents one order of magnitude (×10), a "trip around" the A scale represents two orders of magnitude (×100), and the K scale, three orders of magnitude (×1,000). This relationship allows you to use the A and K scales together with C to find square roots and cube roots, as well as cubes and squares. There is also a special "c" marking on the D scale that helps for calculating the area or diameter of a circle.

4.1. Squares and square roots 🔗

Apply the order-of-magnitude rules to the following if/as needed.

Square root:

  1. Find the original number on the A scale (1–100).
  2. Above that number on the C scale, read the result.
  3. (Note: if you did not add a cursor to your Cardboard computer, use a ruler to line up the numbers with the centre of the wheel.)

Example 1: √150 = ?:

  1. Following the order-of-magnitude rules, divide 150 once by 100 to get 1.5, so that it falls onto the K scale (1–100).
  2. Find 1.5 on the K scale.
  3. Read 1.23 above on the C scale.
  4. Following the order-of-magnitude rules again, since you divided the original number once by 100, multiply the result once by 10 to get the approximate answer 12.3.

Square:

  1. Find the number you want to square on the C scale.
  2. Read the square of the number below on the K scale).
  3. (Note: if you did not add a cursor to your Cardboard computer, use a ruler to line up the numbers with the centre of the wheel.)

Example 2: 7.8² = ?

(Note that you could also simply multiply 7.8 × 7.8 on the C or D scales; however, this alternative approach becomes useful with Combined calculations.)

  1. Find 7.8 on the C scale (it already falls in the range 1–10).
  2. Read the approximate answer 60.1 below on the K scale.

4.2. Cubes and cube roots 🔗

Apply the order-of-magnitude rules to the following if/as needed.

Cube root:

  1. Find the original number on the K scale (1–1,000).
  2. Above that number on the C scale, read the result.
  3. (Note: if you did not add a cursor to your Cardboard computer, use a ruler to line up the numbers with the centre of the wheel.)

Example 1: ∛300

Cube:

  1. Find the original number on the C scale (1–10).
  2. Below that number on the K scale, read the result.
  3. (Note: if you did not add a cursor to your Cardboard computer, use a ruler to line up the numbers with the centre of the wheel.)

Example 2: 8.5³ = ?

5. Combined calculations (advanced only) 🔗

The CI scale is a backwards C scale (think "C Inverted"), running counter-clockwise rather than clockwise. In combination with other scales, it allows you to combine certain operations into a single step.

5.1. Multiplying three numbers 🔗

Apply the order-of-magnitude rules to the following as needed.

  1. Find the first number (factor) on the D scale.
  2. Rotate the wheels until that is directly above the second number on the CI scale.
  3. Without moving the wheels, find the third number on the C scale.
  4. Read the product of the three numbers directly above that on the D scale.

Example 1: 3.5 × 4.8 × 7 = ?

  • Find 3.5 on the D scale.
  • Rotate the wheels until 4.8 is directly below on the CI scale (use the cursor or a ruler to help with alignment).
  • Without moving the wheels, find 7 on the C scale.
  • Read the approximate answer 118 (1.18) directly above on the D scale.

5.2. Circles 🔗

Apply the order-of-magnitude rules to the following as needed.

The special gauge mark c on the D scale represents the ratio between the diameter of a circle and the square root of its area (approximately 1.13). It turns out to be convenient for calculating the area or diameter of a circle.

5.2.1. Area of a circle 🔗
  1. Place the c gauge mark on the D scale directly above the unit pointer (1) on the C scale.
  2. Without moving the wheels, find the diameter of the circle on the D scale.
  3. Read the area of the circle directly below on the A scale (remember that this is a square, so use increments of 100 instead of 10 for orders of magnitude).

Example: calculate the area of a circle with a diameter of 3.7 m

  • Place the c gauge mark on the D scale above the unit pointer (1) on the C scale.
  • Find 3.7 on the D scale.
  • With the help of the cursor or a ruler, read the approximate area 10.7 m² below on the A scale.
5.2.2. Diameter of a circle (from the area) 🔗
  1. Place the c gauge mark on the D scale directly above the unit pointer (1) on the C scale.
  2. Without moving the wheels, find the area of the circle on the A scale (remember that this is a square, so use increments of 100 instead of 10).
  3. Read the diameter of the circle enclosing the area directly below above on the D scale).

Example: calculate the diameter of a circle enclosing an area of 65 m²

  • Place the c gauge mark on the D scale above the unit pointer (1) on the C scale.
  • Find 65 on the A scale.
  • With the help of the cursor or a ruler, read the approximate diameter 9.08 m above on the D scale.

6. Notes 🔗

6.1. Orders of magnitude 🔗

Stop, you might not need to read this!. Most of the time, orders of magnitude won't be a problem: for example, if you divide 75 by 5, it will be obvious that the answer should be 15 rather than 150 or 1.5. Sometimes, however — especially with very large or very small numbers — there's a chance of confusion with the zeros or decimal places. In that case, you can apply the rules in this section to ensure you have the correct result.

Each "trip around the wheel" for the D and C (and CI) scales represents one order of magnitude (10). That means that 2.1 can also represent 21, 210, 2,100, or (in the other direction) .21 or .021. You need to consider the numbers you're multiplying or dividing to decide which order of magnitude applies. For the A scale, each cycle represents two orders of magnitude (100), and for the K scale, each cycle represents three (1,000).

6.1.1. Multiplication magnitude 🔗
  1. Multiply or divide each of the numbers by 10 until they fit into the range 0–10.
  2. Multiply the numbers.
  3. Reverse the operations on the result, dividing by 10 when you multiplied, and multiplying by 10 when you divided.

Example: 15 × 400 = ?

  • Divide 15 by 10 once to get 1.5 (in the range 0–10) and 400 by 10 twice to get 4.
  • Following the instructions in the multiplication section, you'll end up with the result 6.
  • Multiply 6 by 10 three times (reversing the three division operations).
  • The answer is 6,000 (6 × 10 × 10 × 10).
6.1.2. Division magnitude 🔗

Orders of magnitude are a bit trickier with division, because division is not commutative.

  1. Multiply or divide each of the numbers by 10 until they fit into the range 0–10.
  2. Divide the numbers.
  3. On the result, reverse the operations from the first number (dividend), dividing by 10 when you multiplied, and multiplying by 10 when you divided.
  4. Next, repeat the operations from the second number (divisor) multiplying by 10 when you multiplied, and dividing by 10 when you divided.
  5. If the second number (divisor) was originally smaller than the first (dividend), divide once more by 10.

Example 1: 600 ÷ .3 = ?

  • Divide 600 by 10 twice to get 6 (in the range 0–10), and multiply .3 by 10 once to get 3.
  • Following the instructions in the division section, you'll end up with the result 2.
  • Multiply by 10 twice (reversing the division operations on 600), then multiply by 10 once more (repeating the multiplication operation on .03)
  • The answer is 2,000 (2 × 10 × 10 × 10).

Example 2: 30 ÷ 700 = ?

  • Divide 30 by 10 once to get 3 (in the range 0-10), and divide 700 by 10 twice to get 7.
  • Following the instructions in the division section, you'll end up with the result 4.29.
  • Multiply by 10 once (reversion the division operation on 30), then divide by 10 twice (repeating the division operations on 700)
  • Because the dividend 30 is less than divisor 700, divide by 10 once more.
  • The approximate answer is .429 (4.29 × 10 ÷ 10 ÷ 10 ÷ 10).
Square-root magnitude 🔗

With square roots, you divide or multiply the original number on the A scale by 100 (10²) instead of 10, but then still multiply the result on the C scale by 10.

  1. Divide or multiply the original number by 100 (10²) until it falls into the range 1–100 on the A scale,
  2. Perform the square root operation.
  3. Apply the opposite operations to the result from the C scale an order of magnitude lower, dividing by 10 when you originally multiplied by 100, or multiplying by 10 when you originally divided by 100.

Example: √0.003 = ?

  • Multiply by 100 twice to get 30 (in the range 0–100).
  • Following the instructions in the square root section, you'll find 5.48 on the C scale above 30 on the A scale.
  • Divide 5.48 by 10 twice, reversing the two operations on the original number (with 10 rather than 100).
  • The approximate answer is 0.0548 (5.48 ÷ 10 ÷ 10).

(Note: for squares, apply in the opposite order: multiply or divide the original value by 10 until it fits on the C scale, then divide or multiply the result on the A scale by 100 the same number of times.)

Cube-root magnitude 🔗
  1. Divide or multiple the original number by 1,000 (10³) until it falls into the range 1–1,000 on the K scale
  2. Perform the cube root operation.
  3. Apply the opposite operations to the result from the C scale an order of magnitude lower, dividing by 10 when you originally multiplied by 1,000, or multiplying by 10 when you originally divided by 1,000.

Example: ∛2,500 = ?

  • Divide by 1,000 once to get 25 (in the range 0–1,000).
  • Following the instructions in the cube root section, you'll find 5 on the C scale above 25 on the K scale.
  • Multiply 5 by 10, reversing the operation on the original number (but with 10 rather than 1,000).
  • The answer is 50 (5 × 10).

(Note: for cubes, apply in the opposite order: multiply or divide the original value by 10 until it fits on the C scale, then divide or multiply the result on the K scale by 1,000 the same number of times.)

6.2. Precision 🔗

Many answers on the Cardboard Computer will be approximate. The design goal is three significant digits, so you can distinguish between 490 and 491 or 49.0 and 49.1, but not necessarily between 4,900 and 4,901.

High-quality commercial slide rules were often accurate up to 4 or 5 significant digits for experienced users. You might sometimes be able to achieve that level of precision with your own Cardboard Computer, depending on how large you make it and how carefully you align the two wheels, but regardless, you won't get the long sequence of apparently exact digits that you will with a computerised calculator. That's not necessarily a problem: many classic aircraft and even early rockets were designed mostly using analogue slide rules.

6.3. Special gauge marks 🔗

In addition to numbers, the following special gauge marks appear on the C and D scales.

  • π appears just past 3.1 on both the C and D scales, representing the mathematical constant pi (approximately 3.14).
  • c appears just past 1.1 on the D scale only, representing the ratio between the diameter of a circle and the square root of its area (approximately 1.13:1). See 5.2 Circles for more information on using this gauge mark.