Same Math › Geometry · grades 9–10 › Special right triangles
How do special right triangles work?
Two right triangles come up so often that kids learn their side patterns. In a 45-45-90 triangle the legs are equal and the hypotenuse is leg × √2. In a 30-60-90 triangle the hypotenuse is twice the short leg, and the long leg is short leg × √3. Find the short leg (or the leg) first, then multiply.
- 45-45-90: x, x, x√2.
- 30-60-90: x, x√3, 2x.
- The short leg is across from 30°; the hypotenuse is always longest.
Why schools teach it
Exact answers without a calculator
Geometry students connect side ratios to angles (HSG.SRT.C.6) and solve right triangles (HSG.SRT.C.8). These two triangles give exact answers like 5√3 where a calculator would give 8.66.
They come from simple shapes: a square cut on its diagonal makes 45-45-90, and an equilateral triangle cut in half makes 30-60-90. They return in trig and precalculus as the unit circle.
At the kitchen table
What to say, and what to skip
Say this
Which special triangle is it?
Look at the angles.Which side is x?
The leg in 45-45-90; the short leg in 30-60-90.Do you have x, or do you need to divide to get it?
Is the hypotenuse the longest?
Quick check.
Skip this
- Mixing up √2 and √3.Two equal angles, √2. Three different angles, √3.
- Putting x√3 on the hypotenuse.The hypotenuse in 30-60-90 is 2x.
- Not rationalizing.A leg of 6/√2 simplifies to 3√2.
See it in action
A 30-60-90 triangle has a hypotenuse of 14
Hypotenuse is 2x, so find x
2x = 14, x = 7Short leg is x
short leg = 7Long leg is x√3
long leg = 7√3
Check: 7² + (7√3)² = 49 + 147 = 196 = 14² ✓.
Free practice sheet
Make a special right triangles sheet
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Parents also ask
Why √2 and √3?
Pythagoras: x² + x² = 2x² gives x√2; and x² + (x√3)² = 4x² gives 2x.
What if I'm given the long leg of a 30-60-90?
Divide by √3 to get the short leg, then rationalize: 9/√3 = 3√3.
What if I'm given the hypotenuse of a 45-45-90?
Divide by √2: a hypotenuse of 10 gives legs of 10/√2 = 5√2.
Can I use SOH CAH TOA instead?
Yes, but you'll get decimals. Teachers usually want the exact radical answer.