Special Right Triangles Calculator

Calculate all sides of 30-60-90 and 45-45-90 triangles from one known side.

Select Triangle Type

Result

Triangle Type
30-60-90
special right triangle
Short leg (30°)5
Long leg (60°)8.660254
Hypotenuse (90°)10
Area21.650635
Perimeter23.660254

Step-by-Step Solution

30-60-90 ratio: 1 : sqrt(3) : 2

Understanding Special Right Triangles

Special right triangles are right triangles whose angles and side ratios follow specific, predictable patterns. The two most important types are the 30-60-90 triangle and the 45-45-90 triangle. Knowing these ratios allows you to find all sides from just one known side.

Triangle Types and Ratios

30-60-90 Triangle

Angles: 30°, 60°, 90°. The sides are in a fixed ratio based on the shortest side.

Sides: x : x√3 : 2x

45-45-90 Triangle

Angles: 45°, 45°, 90°. An isosceles right triangle with equal legs.

Sides: x : x : x√2

From Short Leg (30-60-90)

If the short leg = a, then long leg = a√3, hypotenuse = 2a.

long = a x sqrt(3), hyp = 2a

From Leg (45-45-90)

If a leg = a, the other leg = a, and the hypotenuse = a√2.

hyp = a x sqrt(2)

Practical Applications

  • Architecture: Roof pitches and structural supports often involve 30-60-90 ratios.
  • Navigation: 45-45-90 triangles appear in diagonal distance calculations.
  • Trigonometry: These triangles provide exact values for sin, cos, and tan of 30°, 45°, and 60°.
  • Standardized Tests: SAT, ACT, and GRE frequently test knowledge of these ratios.