Special Right Triangles: 30-60-90 and 45-45-90 Shortcut Formulas

Certain right-angled triangles appear so frequently in geometry, trigonometry, and standardized tests (SAT, ACT, GRE) that memorizing their side ratios saves significant calculation time.

These are known as Special Right Triangles:

1. The 45° - 45° - 90° Triangle (Isosceles Right Triangle)
2. The 30° - 60° - 90° Triangle

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1. The 45-45-90 Triangle (Isosceles Right Triangle)

A 45° - 45° - 90° triangle is created when a square is cut in half diagonally. Because two internal angles are identical (45°), the two legs are congruent:

Leg a = Leg b = x

Applying the Pythagorean Theorem:

c² = x² + x² = 2x²
c = √(2x²) = x√2

The Ratio:

Leg : Leg : Hypotenuse = 1 : 1 : √2

Rules of Thumb:

• Given a leg x: Multiply by √2 to find the hypotenuse: c = x · √2
• Given the hypotenuse c: Divide by √2 (or multiply by √2 / 2) to find the legs: x = c / √2 = (c · √2) / 2

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2. The 30-60-90 Triangle

A 30° - 60° - 90° triangle is formed by dropping an altitude straight down the middle of an equilateral triangle with side length 2x:

1. The altitude bisects the base into two segments of length x (the shortest leg, opposite 30°).
2. The hypotenuse is the original side length of the equilateral triangle (2x, opposite 90°).
3. The altitude is the longer leg (y, opposite 60°).

Applying the Pythagorean Theorem to find the longer leg:

x² + y² = (2x)²
x² + y² = 4x²
y² = 3x²  ⇒  y = x√3

The Ratio:

Shorter Leg (30°) : Longer Leg (60°) : Hypotenuse (90°) = 1 : √3 : 2

Rules of Thumb:

• Given shorter leg x:
  • Hypotenuse = 2x
  • Longer leg = x√3
• Given hypotenuse c:
  • Shorter leg = c / 2
  • Longer leg = (c · √3) / 2
• Given longer leg y:
  • Shorter leg = y / √3 = (y · √3) / 3
  • Hypotenuse = (2y · √3) / 3

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Comparison Table

Triangle Type	Short Leg	Long Leg	Hypotenuse	Area
45-45-90	x	x	x√2	½ · x²
30-60-90	x	x√3	2x	(√3 / 2) · x²

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Want to verify these triangles with decimal and exact radical answers? Load our preset into the Right Triangle Solver!