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°)

3,4,5 Triangle

Other Lengths

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

Multiples of 3,4,5

Learn more at 3, 4, 5 Triangle

Squaring and Diagonal

Diagonal fixes right angle

How do we ensure two sides are at right angles?

Run a diagonal.

But how long is the diagonal?

calculator x^2

The steps are:

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:

Multiples of 3,4,5


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)
So the diagonal is 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 :

triangle abc

In a right-angled triangle, the square of a (a2) plus the square of b (b2) is equal to the square of c (c2):

a2 + b2 = c2

triangle abc
Take the square root of c2 to get c

So we square a, square b, add those to get c2:

Filling Round Holes

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

circle area is about 80% of square
see circle area for exact values

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

circle area is about 80% of square

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

circle area example

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

circle auger

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 m2
  • 2. Taking 80% of that (estimates a circle): 80% × 0.16 m2 = 0.128 m2
  • 3. And the volume of a 1 m deep hole is: 0.128 m3

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 m3

Estimating Piles

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

cone vs cylinder

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

cone vs cuboid

But be careful: if the base of the heap is much wider in one direction, then this estimate won't work well.