Special Right Triangles

In geometry and trigonometry, there are two "special" right triangles that appear frequently. Memorizing their side ratios allows you to find missing side lengths instantly without relying on the Pythagorean theorem. More importantly, these triangles provide the exact trigonometric values for $30^\circ$, $45^\circ$, and $60^\circ$.

The $45^\circ$-$45^\circ$-$90^\circ$ Triangle

A $45^\circ$-$45^\circ$-$90^\circ$ triangle is an isosceles right triangle. Because the two acute angles are equal, the two legs opposite those angles are also equal in length.

The ratio of the side lengths is $1 : 1 : \sqrt{2}$.

• Legs: $x$
• Hypotenuse: $x\sqrt{2}$

Example: Find the legs of a $45^\circ$-$45^\circ$-$90^\circ$ triangle with a hypotenuse of $10$.

Solution: We know the relationship is $\text{Hypotenuse} = \text{Leg} \cdot \sqrt{2}$. $10 = x\sqrt{2}$ Solving for $x$, we divide by $\sqrt{2}$: $x = \frac{10}{\sqrt{2}}$ Rationalizing the denominator: $x = \frac{10\sqrt{2}}{2} = 5\sqrt{2}$ Both legs have a length of $5\sqrt{2}$.

The $30^\circ$-$60^\circ$-$90^\circ$ Triangle

This triangle is formed by cutting an equilateral triangle perfectly in half.

The ratio of the side lengths is $1 : \sqrt{3} : 2$.

• Short Leg (opposite $30^\circ$): $x$
• Long Leg (opposite $60^\circ$): $x\sqrt{3}$
• Hypotenuse (opposite $90^\circ$): $2x$

Tip: Always find the short leg ($x$) first, as it is the key to finding the other two sides easily.

Example: In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, the side opposite the $30^\circ$ angle is $7$. Find the other sides.

Solution: The side opposite the $30^\circ$ angle is the short leg, so $x = 7$.

• The hypotenuse is twice the short leg: $2x = 2(7) = 14$.
• The long leg (opposite $60^\circ$) is the short leg times $\sqrt{3}$: $x\sqrt{3} = 7\sqrt{3}$.

Exact Trigonometric Values

Because all $45^\circ$-$45^\circ$-$90^\circ$ and $30^\circ$-$60^\circ$-$90^\circ$ triangles are similar, their side ratios give us constant, exact values for sine, cosine, and tangent functions.

For $45^\circ$:

• $\sin(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
• $\cos(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
• $\tan(45^\circ) = \frac{1}{1} = 1$

For $30^\circ$ and $60^\circ$:

• $\sin(30^\circ) = \frac{1}{2}$
• $\cos(30^\circ) = \frac{\sqrt{3}}{2}$
• $\tan(60^\circ) = \frac{\sqrt{3}}{1} = \sqrt{3}$