📐 Similarity & Trigonometry
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Special Right Triangles

Solve 30-60-90 and 45-45-90 triangles using their exact side ratios.

Two right triangle shapes appear so frequently in geometry and trigonometry that their exact side ratios are worth memorizing: the 45-45-90 (isosceles right) triangle and the 30-60-90 triangle.

45-45-90 Triangle

Angles: 45°, 45°, 90°Sides: x, x, x√2Two legs equal; hypotenuse = leg × √2Example: leg = 5 → hypotenuse = 5√2

30-60-90 Triangle

Angles: 30°, 60°, 90°Sides: x, x√3, 2xShort leg x, long leg x√3, hypotenuse 2xExample: short leg = 4 → hyp = 8, long leg = 4√3
45°-45°-90°:  x:x:x230°-60°-90°:  x:x3:2x45°\text{-}45°\text{-}90°:\; x : x : x\sqrt{2} \qquad 30°\text{-}60°\text{-}90°:\; x : x\sqrt{3} : 2x

Solving a Special Right Triangle

1Identify whether the triangle is 45-45-90 or 30-60-90 from the given angles.
2Find which side corresponds to x (short leg), x√3 (long leg), or 2x (hypotenuse).
3Set up an equation and solve for x.
4Find all remaining sides by substituting back into the ratio.
✏️Hypotenuse of 30-60-90
A 30-60-90 triangle has a long leg of 6. Since the long leg = x√3, we get x√3 = 6, so x = 6/√3 = 2√3. The hypotenuse = 2x = 4√3, and the short leg = x = 2√3.
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Remember This!

Connect to trig: sin 45° = cos 45° = 1/√2 = √2/2; sin 30° = 1/2; cos 30° = √3/2; sin 60° = √3/2; cos 60° = 1/2. These come directly from the side ratios.

✏️ Try It!

A 45-45-90 triangle has a hypotenuse of 10. What is the length of each leg?

Practice with CalcVerse
🌊Trig Ratios (SOH‑CAH‑TOA)📐Special Right Triangles
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