Builder's Mathematics
Here are some tips and tricks that may be useful when building.
Need a Right Angle (90°) Fast ... ?
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Make a 3,4,5 Triangle ! Connect three lines:
And we get a right angle (90°) |
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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)
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 (a2) plus the square of b (b2) is equal to the square of c (c2):
a2 + b2 = c2
Add squares of a and b, then square root
So we add the square of a to the square of b, add those to get c2, then take the square root of c2 to get c
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 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
A cone has about one quarter of the volume (closer to 26%) of a surrounding box with a square base:
