3, 4, 5 Triangle

Need a Right Angle (90°) Fast ... ?

Make a 3,4,5 Triangle !

Connect three lines:

  • 3 long
  • 4 long
  • 5 long

And you will have 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

Drawing It

Let us say you need to mark a right angle coming from a point on a wall.

You decide to use 300, 400 and 500 cm lines.

triangle base 300 triangle draw 400 arc triangle draw 500 arc triangle intersect 400 and 500 arc triangle 300 400 500 right angle
Draw a 300 line along the wall Draw an arc 400 away from the start of the 300 line Draw an arc 500 away from the end of the 300 line Connect from the start of the 300 line to where the arcs cross And you have your "3,4,5" triangle with its right angle

The Mathematics Behind It

The Pythagoras Theorem says:

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 3,4,5 with areas 9,15,25
Let's check if it does work:

32 + 42 = 52

Calculating this becomes:

9 + 16 = 25

Yes, it works !

triangle 3,4,5 with areas 9,15,25 lego

Other Combinations

Yes, there are other combinations that work (not just by multiplying). Here are two others:

5,12,13 Triangle   9,40,41 Triangle
5,12,13 triangle   9,40,41 Triangle
52 + 122 = 132   92 + 402 = 412
25 + 144 = 169   (try it yourself)

And there are infinitely many more.... read Pythagorean Triples for more information.

But Wait ... There is More!

The 3,4,5 Triangle has a simple beauty. At its heart is a circle:

3,4,5 Triangle
The circle is a Unit Circle: it has a radius of exactly 1 and an area of π

And using that circle we can cut our triangle into six little triangles like this:

3,4,5 Triangle
Each small triangle is also a right triangle!

And notice that 1+2=3, 1+3=4 and 2+3=5, making the 3,4,5 triangle. So simple and neat.

Is the Radius Really 1?

Imagine we don't know the radius, call it "r" and draw in what we do know:

3,4,5 Triangle

The long side shows us:
5 = (3−r) + (4−r)
Simplify:
5 = 3 + 4 − 2r
Add 2r to both sides:
5 + 2r = 3 + 4
Subtract 5 from both sides
2r = 2
Divide by 2:
r = 1


 

2090,2091