Special Right Triangles

45-45-90 (1:1: 2) and 30-60-90 (1: 3:2).

The idea

Two right triangles occur often enough that their side ratios are worth memorizing: the $45$-$45$-$90$ and the $30$-$60$-$90$ triangle. Their angles are fixed, so their sides always stand in the same ratio, and one known side determines the other two.

Proposition (Special right triangles).

In a $45$-$45$-$90$ triangle the two legs are equal and the sides are in the ratio $1 : 1 : \sqrt{2}$, so the hypotenuse is a leg times $\sqrt{2}$. In a $30$-$60$-$90$ triangle the sides opposite the $30^{\circ}$, $60^{\circ}$ and $90^{\circ}$ angles are in the ratio $1 : \sqrt{3} : 2$, so the long leg is the short leg times $\sqrt{3}$ and the hypotenuse is twice the short leg.

Both ratios come from the Pythagorean theorem. Cut a square along a diagonal. Each half is a right triangle with two equal legs and two equal acute angles of $45^{\circ}$ each — a $45$-$45$-$90$ triangle. Take the legs to be $1$; the Pythagorean theorem gives the hypotenuse $\sqrt{1^{2} + 1^{2}} = \sqrt{2}$. The sides stand in the ratio $1 : 1 : \sqrt{2}$, so the hypotenuse is a leg times $\sqrt{2}$.

Cut an equilateral triangle along an altitude. The cut halves one $60^{\circ}$ angle and meets the base at a right angle, so each half has angles $30^{\circ}$, $60^{\circ}$, $90^{\circ}$ — a $30$-$60$-$90$ triangle. Its hypotenuse is a full side of the equilateral triangle and its short leg is half a side; take these as $2$ and $1$, and the long leg is $\sqrt{2^{2} - 1^{2}} = \sqrt{3}$. The ratio is $1 : \sqrt{3} : 2$.

Only the ratio is fixed, not the size, so scale it to the given side.

Ways to work on it

• Walkthrough. The side ratios of the two special right triangles.
• Practice. One special triangle, one missing side.
• Hardest. Work backwards from a hypotenuse or long leg.

Not sure where to start? Take the ten-question placement test.