Special right triangles are right triangles with angle measures that create predictable side ratios. The two most useful types are 30°-60°-90° triangles and 45°-45°-90° triangles. Because their side lengths follow fixed patterns, you can find missing sides quickly without a calculator.
These triangles appear often in geometry, trigonometry, physics, engineering, and design.
Key Facts
- In a 45°-45°-90° triangle, the side ratio is leg : leg : hypotenuse = x : x : x√2.
- In a 30°-60°-90° triangle, the side ratio is short leg : long leg : hypotenuse = x : x√3 : 2x.
- The hypotenuse is always the longest side and is always opposite the 90° angle.
- In a 30°-60°-90° triangle, the short leg is opposite 30° and the long leg is opposite 60°.
- In a 45°-45°-90° triangle, the two legs are congruent because the acute angles are equal.
- To scale a special right triangle, multiply every part of the ratio by the same number.
Vocabulary
- Hypotenuse
- The hypotenuse is the side opposite the right angle and is the longest side of a right triangle.
- Leg
- A leg is one of the two sides that form the right angle in a right triangle.
- 30°-60°-90° triangle
- A 30°-60°-90° triangle is a right triangle whose side lengths follow the ratio x : x√3 : 2x.
- 45°-45°-90° triangle
- A 45°-45°-90° triangle is an isosceles right triangle whose side lengths follow the ratio x : x : x√2.
- Side ratio
- A side ratio compares the side lengths of similar triangles using proportional values.
Common Mistakes to Avoid
- Mixing up the long leg and hypotenuse in a 30°-60°-90° triangle is wrong because the hypotenuse is 2x, not x√3.
- Putting x√2 on a leg of a 45°-45°-90° triangle is wrong because x√2 is the hypotenuse and both legs are x.
- Forgetting that the short leg is opposite 30° is wrong because side lengths are matched to the angles across from them.
- Multiplying only one side by a scale factor is wrong because similar triangles require all corresponding sides to be scaled by the same factor.
Practice Questions
- 1 A 45°-45°-90° triangle has legs of length 6 cm. Find the length of the hypotenuse.
- 2 A 30°-60°-90° triangle has a short leg of length 5 m. Find the long leg and the hypotenuse.
- 3 A student says the side opposite 60° in a 30°-60°-90° triangle is the hypotenuse. Explain why this is incorrect and identify the correct side.