Calculate 30-60-90 triangle short leg 3 m Step by Step

Step by step

The question

Question: find the other two sides of a 30-60-90 triangle with a short leg of 3 m.

1 Half of an equilateral

The angles are 30°, 60° and 90°.

Put a copy under it, upside down.

Together they make a triangle with three equal sides.

That is an equilateral triangle.

All three sides are equal, and every angle is 60°.

The short leg is across from the 30° angle.

It is half of one side.

We call it x.

The other half of that side is x too.

The hypotenuse is one whole side.

So it is 2 times the short leg.

The long leg is the short leg times √3.

2 Find the short leg

The short leg is given: x = 3 m.

x = 3 m

3 Find the hypotenuse

The hypotenuse is 2x: 2 times 3.

hypotenuse = 2 × 3

So the hypotenuse is 6 m.

hypotenuse = 6 m

4 Find the long leg

The long leg is x times √3: 3 times √3.

long leg = 3 × √3

Why √3?

Pythagoras: the hypotenuse squared, 4x², take away x².

That leaves 3x².

Its square root is x times √3.

(2x)² − x² = 3x² → x√3

So the long leg is 3√3 m.

long leg = 3√3 m

Careful!

√3 is for the long leg of this triangle.

√2 is for the 45-45-90 triangle.

Careful: not 3√2 ✗

So the hypotenuse is 6 m and the long leg is 3√3 m.

hypotenuse 6 m and long leg 3√3 m

5 Check

Let's check with Pythagoras: short leg squared plus long leg squared.

3² + (3√3)² = 9 + 9 × 3 = 9 + 27 = 36

The hypotenuse squared: 6² is 36.

6² = 36

Both are 36.

So the sides are right!

Let's look back.

First: half of an equilateral.

Then: find the short leg.

Then: find the hypotenuse.

Then: find the long leg.

Last: check.

Answer