30-60-90 — 30-60-90 triangle side ratios · Principia

Dependencies: Right triangle: median to hypotenuse = half hypotenuse (D.08), Base angles of an isoceles triangle are equal (C.01), Isoceles with a 60° angle ⇒ equilateral (C.04).

Statement

Let right $\triangle ABC$ have interior angles $\angle A = 30^\circ$ , $\angle B = 60^\circ$ , $\angle C = 90^\circ$ . Then the leg opposite the $30^\circ$ angle is exactly half the hypotenuse:

|BC| \;=\; \tfrac{1}{2}\,|AB|.

Plugging in the other leg, the side ratio falls out immediately:

|BC| : |AC| : |AB| \;=\; 1 : \sqrt{3} : 2.

$30°-60°-90° \triangle ABC with the three angles and the side ratio 1:\sqrt{3}:2.$

30°-60°-90° right triangle: side opposite 30° = hypotenuse / 2

Statement

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