How this calculator works
Special triangles have fixed side ratios. A known length or perimeter sets a linear scale; a known area sets the square of that scale. The 30°-60°-90° ratio comes from halving an equilateral triangle, while the 45°-45°-90° ratio comes from bisecting a square. Integer triples provide three further shapes with exact rational side ratios.
Formula
30°-60°-90°: a:b:c = 1:√3:2. 45°-45°-90°: 1:1:√2. At scale k, K = k²a₀b₀/2 and P = k(a₀+b₀+c₀).
Worked example
For a 30°-60°-90° triangle with short leg 2 cm, the other leg is 2√3 cm and the hypotenuse is 4 cm. A 45°-45°-90° triangle with hypotenuse 10 m has legs 5√2 m and area 25 m².
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Frequently asked questions
Which leg is opposite 30°?
The short leg a is opposite 30°. The long leg b is opposite 60°.
Are the two 45° legs interchangeable?
Yes. They are equal, so either can be entered as a or b.
Will scaling preserve the angles?
Yes. Multiplying every side by the same positive number preserves all angles and multiplies area by the square of that number.