Special Right Triangles: 45-45-90 and 30-60-90

Two triangles worth knowing exactly

A calculator gives $\sin 45° \approx 0.7071$. That is a rounded decimal standing in for an exact value:

$$\sin 45° = \frac{\sqrt{2}}{2}$$

Two right triangles have side ratios exact enough to write down, and both come from cutting a familiar figure in half. They appear so often that recognizing them saves the calculator step entirely.

45-45-90: half a square

Cut a square along its diagonal. The two halves are right triangles, each with two $45°$ angles.

The legs are the sides of the square, so they are equal. Call each one $1$ and use the Pythagorean theorem:

$$c^2 = 1^2 + 1^2 = 2 \quad\Rightarrow\quad c = \sqrt{2}$$ $$1 : 1 : \sqrt{2}$$

The hypotenuse is a leg times $\sqrt{2}$. Going the other way, a leg is the hypotenuse divided by $\sqrt{2}$.

Leg	Hypotenuse
$1$	$\sqrt{2}$
$5$	$5\sqrt{2}$
$\tfrac{7}{\sqrt{2}}$	$7$

30-60-90: half an equilateral triangle

Take an equilateral triangle with sides of $2$ and drop a perpendicular from one vertex to the opposite side.

That line bisects the base and bisects the top angle. Each half is a right triangle with angles $30°$, $60°$ and $90°$.

The hypotenuse is a full side, $2$. The short leg is half the base, $1$. The Pythagorean theorem gives the third:

$$h^2 = 2^2 - 1^2 = 3 \quad\Rightarrow\quad h = \sqrt{3}$$ $$1 : \sqrt{3} : 2$$

Which side is which follows from a rule that always holds: the largest angle faces the longest side.

Opposite the	Side
$30°$	$1$ — the short leg
$60°$	$\sqrt{3} \approx 1.73$
$90°$	$2$ — the hypotenuse

The single most useful fact here is that the hypotenuse is twice the short leg. Everything else can be rebuilt from that plus Pythagoras.

Why the ratios can be scaled freely

All 45-45-90 triangles are similar to each other, because they share all three angles. Same for all 30-60-90 triangles.

So the ratios above hold at every size, and a triangle with a known side is found by scaling:

A 30-60-90 triangle has a short leg of $7$. Find the other sides.

$$\text{hypotenuse} = 2 \times 7 = 14 \qquad \text{long leg} = 7\sqrt{3}$$

Multiply the whole ratio by whatever makes the known side match.

Identify which side you were given first. Being handed the long leg rather than the short one changes every step, and it is the error this topic produces.

The exact trig values

Reading the ratios off these two triangles gives the values usually memorized:

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$
$30°$	$\tfrac{1}{2}$	$\tfrac{\sqrt{3}}{2}$	$\tfrac{1}{\sqrt{3}}$
$45°$	$\tfrac{\sqrt{2}}{2}$	$\tfrac{\sqrt{2}}{2}$	$1$
$60°$	$\tfrac{\sqrt{3}}{2}$	$\tfrac{1}{2}$	$\sqrt{3}$

Two patterns are worth noticing rather than memorizing.

The $30°$ and $60°$ columns are mirror images, which is the co-function rule: $\sin 30° = \cos 60°$.

And $\tan 45° = 1$ because the legs are equal, so opposite over adjacent is one.

$\sin 30° = \tfrac{1}{2}$ says something concrete: the side opposite a $30°$ angle is exactly half the hypotenuse. That single sentence is the whole of the 30-60-90 triangle in words.

Worked examples

Common mistakes

Practice problems