Overview

These two ratios show up everywhere. Memorize them to avoid re-deriving, and remember how they come from symmetry and equilateral triangles.

Key Ideas

• 45-45-90: $1:1:\sqrt{2}$.
• 30-60-90: $1:\sqrt{3}:2$.
• A 45-45-90 triangle comes from cutting a square along its diagonal.
• A 30-60-90 triangle comes from splitting an equilateral triangle in half.

Core Skills

Choose the Reference Side

Identify the short leg in a 30-60-90 triangle or a leg in a 45-45-90 triangle, then scale the entire ratio.

Use Symmetry

If a triangle has two equal sides or angles, consider dropping an altitude to create a special right triangle.

Combine with Pythagorean Theorem

If a side is missing, decide whether the ratio or $a^2+b^2=c^2$ is faster.

Worked Example

A 30-60-90 triangle has hypotenuse $8$. Find the longer leg.

The short leg is $4$, so the long leg is $4\sqrt{3}$.

More Examples

Example 1: 45-45-90

The diagonal of a square is $10$. Find the side length.

In a 45-45-90 triangle, $c = s\sqrt{2}$, so $s = 10/\sqrt{2} = 5\sqrt{2}$.

Example 2: 30-60-90 from Equilateral

An equilateral triangle has side $12$. Find the altitude.

The altitude splits it into two 30-60-90 triangles, so the altitude is $6\sqrt{3}$.

Strategy Checklist

• Identify which angle is $30^\circ$ or $45^\circ$ before scaling.
• Scale all three sides together.
• Use these ratios to avoid unnecessary square roots.

Common Pitfalls

• Mixing which side corresponds to $\sqrt{3}$ or $2$.
• Forgetting to scale all sides together.
• Using the short leg when the given side is actually the long leg.

Practice Problems