Special Right Triangles
Two triangles with memorizable side ratios — 45-45-90 and 30-60-90. Learn the ratios once, then read every missing side instantly in exact radical form.
1 The two ratios
Both come straight from the Pythagorean theorem — but you never need to use it again.
45-45-90 leg : leg : hypotenuse = 1 : 1 : √2
The legs are equal. The hypotenuse is always leg × √2.
Leg 5 → hypotenuse 5√2. Hypotenuse 8√2 → leg 8.
30-60-90 short : long : hyp = 1 : √3 : 2
The short leg sits opposite 30°; the long leg (×√3) opposite 60°; hypotenuse is 2 × short.
Short 4 → long 4√3, hyp 8.
The #1 mix-up: in 30-60-90, the leg opposite 30° is the short one (1),
and the leg opposite 60° is the long one (√3). Say it with the angles: “thirty is short.”
Keep it exact: answers like 5√2 are preferred over decimals. Never leave a radical
in a denominator — rationalize: 10/√2 = 5√2 and 6/√3 = 2√3.
Quick check 1 — 45-45-90 triangle, leg = 5. The hypotenuse is…
Quick check 2 — 45-45-90 triangle, hypotenuse = 8√2. A leg is…
Quick check 3 — 30-60-90 triangle, short leg = 4. The long leg is…
Quick check 4 — 30-60-90 triangle, short leg = 4. The hypotenuse is…
2 Worked examples
Going backwards — and rationalizing the denominator.
Example 1 — rationalize
45-45-90 triangle, hypotenuse = 10. Find a leg.
- Hypotenuse = leg × √2, so leg = 10/√2.
- Rationalize: multiply top and bottom by √2: 10√2/2.
- Simplify: 5√2.
Example 2 — backwards in 30-60-90
30-60-90 triangle, hypotenuse = 10. Find the short leg.
- Hypotenuse = 2 × short leg, so short = 10/2 = 5.
- (The long leg would be 5√3.)
Quick check 5 — 30-60-90 triangle, long leg = 6√3. The short leg is…
Quick check 6 — In a 30-60-90 triangle, the side opposite the 30° angle is the…