Special Right Triangles: 30-60-90 and 45-45-90

An interactive exploration of the two special right triangles and their fixed side ratios: the 30-60-90 triangle ($1 : \sqrt{3} : 2$) derived from an equilateral triangle, and the 45-45-90 triangle ($1 : 1 : \sqrt{2}$) from an isosceles right triangle.

Keywords: special right triangles, 30-60-90, 45-45-90, equilateral triangle, isosceles right triangle, side ratios

Prerequisites: pythagorean-theorem, angle-sum-proof · Difficulty: intermediate

Two special right triangles appear so often in geometry and trigonometry that their side ratios are worth memorizing.

The 30-60-90 Triangle

Start with an equilateral triangle (all sides $2s$, all angles $60°$). Drop an altitude from one vertex to the opposite side. The altitude bisects the base and the vertex angle, creating two congruent 30-60-90 triangles with sides:

s \;:\; s\sqrt{3} \;:\; 2s
    

(short leg : long leg : hypotenuse)

The 45-45-90 Triangle

Take a square and cut along the diagonal, or simply consider an isosceles right triangle with two equal legs of length $s$. By the Pythagorean theorem the hypotenuse is $s\sqrt{2}$:

s \;:\; s \;:\; s\sqrt{2}
    

Why They Matter

These ratios let you find missing sides without a calculator whenever the angles are $30°$, $45°$, $60°$, or $90°$.

The Equilateral Triangle

We begin with an equilateral triangle $PQR$ where every side has length $2s$ and every angle is $60°$.

Drop the Altitude

Drop the altitude from $P$ to side $QR$. Because the triangle is equilateral, the altitude bisects $QR$ at point $F$, so $QF = FR = s$. The altitude also bisects angle $P$ into two $30°$ angles.

30-60-90 Side Ratios

In the 30-60-90 triangle $PFR$:

• Short leg $FR = s$ (half the equilateral side, opposite $30°$)
• Hypotenuse $PR = 2s$ (a full side of the equilateral triangle)
• Long leg $PF = s\sqrt{3}$ (by the Pythagorean theorem)

The ratio is $s : s\sqrt{3} : 2s$, or simply $1 : \sqrt{3} : 2$.

The Isosceles Right Triangle

Now consider isosceles right triangle $XYZ$ with the right angle at $Y$. The two legs $XY$ and $YZ$ are equal (both length $s$), and the two base angles are each $45°$.

45-45-90 Side Ratios

In the 45-45-90 triangle:

• Both legs equal $s$
• Hypotenuse $= \sqrt{s^2 + s^2} = s\sqrt{2}$

The ratio is $s : s : s\sqrt{2}$, or simply $1 : 1 : \sqrt{2}$.

Together with the 30-60-90 triangle, these two special right triangles let you solve any problem involving $30°$, $45°$, $60°$, or $90°$ angles without a calculator.

Conclusion

The two special right triangles have fixed, memorable side ratios:

• The 30-60-90 comes from splitting an equilateral triangle.
• The 45-45-90 is the isosceles right triangle.