# Builder's Mathematics

Here are some tips and tricks that may be useful when building.

## Need a Right Angle (90°) Fast ... ?

Make a 3,4,5 Triangle ! Connect three lines: - 3 long
- 4 long
- 5 long
And we get a right angle (90°) |

### Other Lengths

You can use other lengths by multiplying each side by 2, or by 10, or any multiple:

Learn more at 3, 4, 5 Triangle

## Squaring and Diagonal

How do we ensure two sides are at right angles?

**Run a diagonal**.

But how long is the diagonal?

The steps are:

- measure side
**a**and square it (multiply it by itself) - measure side
**b**and square it also - add those squares
- finish with square root

### Example: A frame with sides of 2.4 and 5.365

- 2.4 squared is 2.4×2.4 =
**5.76** - 5.365 squared is 5.365×5.365 =
**28.783225** - 5.76 + 28.783225 =
**34.543225** - square root of 34.543225 is
**5.877**(rounded to 3 decimal places)

And we get this:

Perfection!

### Example: Sides are 300 and 450.5

- 300 squared is 300×300 =
**90000** - 450.5 squared is 450.5×450.5 =
**202950.25** - 90000 + 202950.25 =
**292950.25** - square root of 292950.25 is
**541.25**(rounded to 2 decimal places)

**541.25**

Notice how the squares can get very big, but come back to normal when we do the square root at the end

Try a few values here:

Note: it is easy to slip a digit when doing these calcs, so double check!

Why does it work? It is Pythagoras' Theorem :

In a right-angled triangle, the square of a (a^{2}) plus the square of b (b^{2}) is equal to the square of c (c^{2}):

a^{2} + b^{2} = c^{2}

Take the square root of c^{2} to get **c**

So we square a, square b, add those to get c^{2}:

## Filling Round Holes

A circle has **about 80%** of the area of a similar-width square:

see circle area for exact values

So a circular hole has **about 80%** of the volume of a squared-off hole!

### Example: You want to drill foundation holes and fill them with concrete.

The holes are **0.4 m wide** and **1 m deep**, how much concrete should you order for each hole?

They are circular (in cross section) because they are drilled out using an auger.

You can make an **estimate** by:

- 1. Calculating a square hole: 0.4 × 0.4 =
**0.16 m**^{2} - 2. Taking 80% of that (estimates a circle): 80% × 0.16 m
^{2}=**0.128 m**^{2} - 3. And the volume of a 1 m deep hole is:
**0.128 m**^{3}

So you should order 0.128 cubic meters of concrete to fill each hole.

Note: a more accurate calculation using the circle's true area gives **0.126 m ^{3}**

## Estimating Piles

A cone (such as a heaped pile of sand) **has exactly one third** of the volume of a surrounding cylinder

A cone has **about one quarter** of the volume (closer to 26%) of a surrounding box with a square base: