Calculate 45-45-90 triangle leg 3

Step by step

The question

Question: find the hypotenuse of a 45-45-90 triangle with a leg of 3.

1 Half of a square

The angles are 45°, 45° and 90°.

This is half of a square.

The longest side cuts the square in two.

So the two legs are sides of the square.

They are equal: both are 3.

2 Use the pattern

The pattern: the hypotenuse is a leg times √2.

Here the leg is 3.

So c = 3 × √2.

Why √2?

Pythagoras: x² + x² is 2x².

The square root of 2x² is x times √2.

We leave out the times sign.

So c = 3√2.

That is about 4.2.

Careful!

The hypotenuse is not the two legs added.

The way along the diagonal is shorter than going round two sides.

3 Check

Let's check with Pythagoras: a² + b².

Now c²: (3√2)² is 3² times 2.

Both are 18.

So c = 3√2 is right!

Let's look back.

First: half of a square.

Then: use the pattern.

Last: check.