Special Right Triangles: 45-45-90 and 30-60-90
Derive the exact side ratios for 45-45-90 (1:1:√2) and 30-60-90 (1:√3:2) triangles. Memorize these ratios and apply them to find exact side lengths without a calculator, essential for standardized tests and proofs.
A lesson of Geometry, in Right Triangles and Trigonometry, about 10 minutes.
By the end of this lesson, you will be able to find all side lengths of 45-45-90 and 30-60-90 triangles given one side, apply these patterns without a calculator in timed settings, and recognize when a problem uses a special right triangle.
The Big Idea
Two right triangle configurations appear constantly in mathematics, physics, and engineering: the 45-45-90 (half a square) and 30-60-90 (half an equilateral triangle). Their side ratios are fixed and derivable from the Pythagorean theorem. Memorizing these patterns lets you solve a large class of problems instantly.
45-45-90 Triangle
Angles: 45°, 45°, 90°
Origin: Cut a square along a diagonal. The legs are the sides of the square; the diagonal is the hypotenuse.
Side ratios: If each leg = $x$:
$$\text{hypotenuse} = x\sqrt{2}$$
| Leg | Leg | Hypotenuse |
|---|---|---|
| $x$ | $x$ | $x\sqrt{2}$ |
| $1$ | $1$ | $\sqrt{2}$ |
| $3$ | $3$ | $3\sqrt{2}$ |
| $5$ | $5$ | $5\sqrt{2}$ |
Derivation: $c = \sqrt{x^2 + x^2} = \sqrt{2x^2} = x\sqrt{2}$
30-60-90 Triangle
Angles: 30°, 60°, 90°
Origin: Cut an equilateral triangle in half with an altitude. The short leg is half the original side.
Side ratios: If the short leg = $x$:
| Short leg (opp. 30°) | Long leg (opp. 60°) | Hypotenuse (opp. 90°) |
|---|---|---|
| $x$ | $x\sqrt{3}$ | $2x$ |
| $1$ | $\sqrt{3}$ | $2$ |
| $5$ | $5\sqrt{3}$ | $10$ |
Derivation: Equilateral triangle with side $2x$, altitude splits into two 30-60-90 triangles with hypotenuse $2x$ and short leg $x$. Long leg: $\sqrt{(2x)^2 - x^2} = \sqrt{3x^2} = x\sqrt{3}$.