Key takeaway: Special right triangles are the 45-45-90 and 30-60-90 shapes, whose side lengths follow fixed ratios, so every missing side can be found from one known side without a calculator.
Grade 10 – GeometryG-SRT.C.6Trigonometry
Definition of Special Right Triangles
Two shapes of right triangle show up so often that their side ratios are worth memorizing, and they are called special right triangles.
The first is half of a square cut along its diagonal: both legs match and the hypotenuse is a leg times \(\sqrt{2}\). The second is half of an equilateral triangle sliced along its altitude: its slanted side doubles the short leg, while the remaining leg equals the short leg multiplied by \(\sqrt{3}\). The short leg faces the \(30^{\circ}\) corner.
Formula and notation
For special right triangles, let \(x\) be the smaller leg (or either leg of the isosceles one):
\[ 45\text{-}45\text{-}90:\ \ x,\ x,\ x\sqrt{2} \qquad\qquad 30\text{-}60\text{-}90:\ \ x,\ x\sqrt{3},\ 2x \]
The sides are listed in order from the smallest angle to the largest.
Key facts
- In the isosceles shape the legs match and the hypotenuse is about 1.414 times a leg.
- In the half-equilateral shape the hypotenuse is exactly double the shortest side.
- The shortest side always faces the smallest angle.
- These ratios let you skip sine, cosine and tangent for 30, 45 and 60 degree angles.
Special Right Triangles example
A shelf bracket is a right triangle with angles of \(30^{\circ}\) and \(60^{\circ}\) and a slanted side of 38 cm. Find both legs. A second bracket has two \(45^{\circ}\) angles and legs of 11 cm. Find its slanted side.
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First bracket: the slanted side is \(2x = 38\), so the short leg is \(x = 19\) cm.
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The long leg is \(x\sqrt{3} = 19\sqrt{3} \approx 32.9\) cm.
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Second bracket: the slanted side is \(11\sqrt{2} \approx 15.6\) cm.
The first bracket has legs 19 cm and \(19\sqrt{3} \approx 32.9\) cm; the second has a slanted side of \(11\sqrt{2} \approx 15.6\) cm.
Common mistake
Watch out
Gabe says the slanted side of the half-equilateral bracket with short leg 19 is \(19\sqrt{3}\), about 32.9 cm.
Why it is wrong
He used the long leg’s multiplier for the hypotenuse. The slanted side is \(2x = 38\); \(19\sqrt{3}\) is the leg facing \(60^{\circ}\). The longest side must face the right angle, and 38 is bigger than 32.9.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Trigonometry