Special Right Triangles (30-60-90 and 45-45-90) Cheat Sheet

A printable reference covering 30-60-90 ratios, 45-45-90 ratios, hypotenuse relationships, leg relationships, and exact radical lengths for grades 8-11.

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Special right triangles are right triangles with angle measures that create predictable side length ratios. This cheat sheet covers the two most important types, $30^\circ$ - $60^\circ$ - $90^\circ$ and $45^\circ$ - $45^\circ$ - $90^\circ$ triangles. Students use these patterns to find missing side lengths quickly without needing a calculator. These triangles also appear often in geometry proofs, trigonometry, coordinate geometry, and standardized tests. In a $45^\circ$ - $45^\circ$ - $90^\circ$ triangle, the legs are equal and the hypotenuse is $\sqrt{2}$ times a leg. In a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle, the short leg, long leg, and hypotenuse follow the ratio $1:\sqrt{3}:2$ . The shortest side is always across from $30^\circ$ , and the longest side is always the hypotenuse. Knowing which side you are given helps you multiply or divide by $\sqrt{2}$ , $\sqrt{3}$ , or $2$ correctly.

Key Facts

In a $45^\circ$ - $45^\circ$ - $90^\circ$ triangle, the side ratio is $x:x:x\sqrt{2}$ .
In a $45^\circ$ - $45^\circ$ - $90^\circ$ triangle, if each leg is $x$ , then the hypotenuse is $x\sqrt{2}$ .
In a $45^\circ$ - $45^\circ$ - $90^\circ$ triangle, if the hypotenuse is $h$ , then each leg is $\frac{h}{\sqrt{2}}=\frac{h\sqrt{2}}{2}$ .
In a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle, the side ratio is $x:x\sqrt{3}:2x$ .
In a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle, the short leg $x$ is across from $30^\circ$ .
In a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle, the long leg $x\sqrt{3}$ is across from $60^\circ$ .
In a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle, the hypotenuse $2x$ is twice the short leg.
The Pythagorean Theorem $a^2+b^2=c^2$ confirms both special right triangle patterns.

Vocabulary

Right Triangle: A triangle with one angle measuring $90^\circ$ .
Hypotenuse: The longest side of a right triangle, located across from the $90^\circ$ angle.
Leg: Either of the two sides that form the right angle in a right triangle.
45-45-90 Triangle: An isosceles right triangle with angles $45^\circ$ , $45^\circ$ , and $90^\circ$ and side ratio $x:x:x\sqrt{2}$ .
30-60-90 Triangle: A right triangle with angles $30^\circ$ , $60^\circ$ , and $90^\circ$ and side ratio $x:x\sqrt{3}:2x$ .
Radical Form: An exact form using a square root, such as $6\sqrt{3}$ , instead of a rounded decimal.

Common Mistakes to Avoid

Using the same ratio for both triangle types is wrong because $45^\circ$ - $45^\circ$ - $90^\circ$ triangles use $x:x:x\sqrt{2}$ , while $30^\circ$ - $60^\circ$ - $90^\circ$ triangles use $x:x\sqrt{3}:2x$ .
Putting the short leg across from $60^\circ$ is wrong because in a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle, the short leg $x$ is always across from $30^\circ$ .
Multiplying by $\sqrt{2}$ to find a leg from the hypotenuse in a $45^\circ$ - $45^\circ$ - $90^\circ$ triangle is wrong because you should divide: $x=\frac{h}{\sqrt{2}}$ .
Forgetting that the hypotenuse is the longest side is wrong because neither leg can be longer than the side across from the $90^\circ$ angle.
Rounding radical answers too early is wrong because exact answers such as $4\sqrt{3}$ or $7\sqrt{2}$ are usually required in geometry.

Practice Questions

1 A $45^\circ$ - $45^\circ$ - $90^\circ$ triangle has legs of length $6$ . Find the hypotenuse.
2 A $30^\circ$ - $60^\circ$ - $90^\circ$ triangle has a short leg of length $5$ . Find the long leg and hypotenuse.
3 A $45^\circ$ - $45^\circ$ - $90^\circ$ triangle has a hypotenuse of $10\sqrt{2}$ . Find the length of each leg.
4 Explain how you can decide whether to use the ratio $x:x:x\sqrt{2}$ or $x:x\sqrt{3}:2x$ when solving a special right triangle.

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