Special Right Triangles Calculator

The formula

sides are in the ratio 1 : √3 : 2

Why 1 : √3 : 2

Cut an equilateral triangle in half down its height and you get a 30-60-90. The hypotenuse is a full side, the short leg is half a side, and the long leg is the height — which Pythagoras makes √(1² − 0.5²) = √3/2 of a side. Doubling through gives the familiar 1 : √3 : 2.

What always holds

• The three angles add to 180°, without exception.
• Any two sides must add to more than the third — the triangle inequality. Sides of 4, 5 and 12 cannot close.
• The longest side always faces the largest angle.
• Area = ½ × base × height for any triangle, with the height measured perpendicular to that base.

Which formula to reach for

With three sides, use Heron’s formula. With a base and its height, use ½bh. With two sides and the angle between them, use ½ab·sin C. With a right angle, the two legs are already base and height, so the area is just half their product.