Math July 21, 2026 · 11 min read

Special Right Triangles Calculator: 30-60-90 and 45-45-90 Guide (2026)

Calculate all sides of special right triangles — 30-60-90 and 45-45-90. Enter one side to find all remaining sides instantly.

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Put these formulas into practice with our instant, step-by-step Special Right Triangles Calculator.

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Special right triangles have fixed angle and side ratios that make them incredibly useful in geometry, trigonometry, and real-world applications. The two most important types are the 30-60-90 triangle (sides in ratio 1:√3:2) and the 45-45-90 triangle (sides in ratio 1:1:√2). Knowing these ratios allows you to find all sides from a single known side without a calculator.

Key Takeaway

In a 30-60-90 triangle, the sides are 1:√3:2 (shortest:longer:hypotenuse). In a 45-45-90 triangle, the sides are 1:1:√2 (leg:leg:hypotenuse). Enter any side to find all others.

30-60-90 Triangle Ratios

Shortest leg (30°) : Longer leg (60°) : Hypotenuse (90°) = 1 : √3 : 2

The 30-60-90 triangle is half of an equilateral triangle. If you draw the altitude of an equilateral triangle with side 2a, it splits into two 30-60-90 triangles with sides a, a√3, and 2a. The shortest leg is opposite the 30° angle, the longer leg is opposite the 60° angle, and the hypotenuse is opposite the 90° angle.

45-45-90 Triangle Ratios

Leg : Leg : Hypotenuse = 1 : 1 : √2

The 45-45-90 triangle is half of a square. Drawing the diagonal of a square with side s creates two 45-45-90 triangles with legs s and hypotenuse s√2. Both legs are equal because both acute angles are 45°.

How to Use the Calculator

  1. Select the triangle type: 30-60-90 or 45-45-90.
  2. Enter the known side length.
  3. Indicate which side you entered.
  4. The calculator shows all three sides with step-by-step derivations.

Frequently Asked Questions

Where do 30-60-90 triangles come from?
They are half of an equilateral triangle, created by drawing the altitude.
Where do 45-45-90 triangles come from?
They are half of a square, created by drawing the diagonal.
How are these used in trigonometry?
They give exact values: sin(30°) = 1/2, cos(45°) = √2/2, tan(60°) = √3, etc.

Conclusion

Special right triangles are foundational to geometry and trigonometry. Use our free special right triangles calculator to find all sides from any known dimension.