A 30-60-90 triangle is a right triangle whose angles measure 30°, 60° and 90°. Its sides are always in the ratio 1 : √3 : 2: if the short leg (opposite 30°) is x, the long leg is x√3 and the hypotenuse is 2x. So knowing one side gives the other two.
In Spanish
triángulo 30-60-90
Also called
30 60 90 triangle, 30°-60°-90° triangle, special right triangle
Half of an equilateral triangle: the short leg is half the side, so the hypotenuse is twice the short leg, and the Pythagorean theorem gives the long leg x√3.
Formal definition and notation
A 30-60-90 triangle is a triangle with interior angles of 30°, 60° and 90°. Cutting an equilateral triangle of side 2x along an altitude gives two of them, so the side opposite 30° is x, the hypotenuse is 2x, and by the Pythagorean theorem the side opposite 60° is √(2x)2 − x2 = x√3. All 30-60-90 triangles are similar, which gives exact trigonometric values: sin 30° = 12, cos 30° = √32, tan 30° = √33.
Symbol
Read it as
Meaning
x : x√3 : 2x
x to x root 3 to 2x
short leg : long leg : hypotenuse
Key facts
The hypotenuse is exactly twice the short leg: hypotenuse 10 means short leg 5.
The long leg is about 1.732 times the short leg, because √3 ≈ 1.732.
Matching sides to angles: the smallest side faces the smallest angle (30°), and the largest side faces 90°.
An equilateral triangle with side s has heights√32, because the height is the long leg of a 30-60-90 triangle with hypotenuse s.
“Special right triangles” is the family name for both the 30-60-90 and the 45-45-90 triangles; this page covers only the 30-60-90 rules.
30-60-90 Trianglevs.3-4-5 triangle
A 3-4-5 triangle is a right triangle with whole-number sides, but its acute angles are about 36.9° and 53.1°, not 30° and 60°.
Quick check: 3 questions
Question 1 of 3
The short leg of a 30-60-90 triangle is 4. What is the hypotenuse?
Show answer and explanation
Answer: B. Hypotenuse = 2 × short leg = 8. 4√3 is the long leg, 4√2 is a 45-45-90 hypotenuse, and 6 adds 2 instead of doubling.
Question 2 of 3
The hypotenuse of a 30-60-90 triangle is 10. How long is the long leg?
Show answer and explanation
Answer: C. Short leg = 10 ÷ 2 = 5, so long leg = 5√3. 5 is the short leg, 10√3 skips halving, and 5√2 uses the 45-45-90 rule.
Question 3 of 3
In a 30-60-90 triangle the side opposite 30° is 6. What is the side opposite 60°?
Show answer and explanation
Answer: D. Opposite 60° is the long leg: 6√3. 12 is the hypotenuse, 3√3 divides instead of multiplying, and 6√2 uses the wrong ratio.
30-60-90 Triangle: frequently asked questions
What is the 30-60-90 triangle rule?
The sides are in the ratio1 : √3 : 2. Call the side opposite the 30° angle x; then the side opposite 60° is x√3 and the hypotenuse is 2x. With a short leg of 7, the long leg is 7√3 ≈ 12.1 and the hypotenuse is 14.
Why is the hypotenuse twice the short leg?
Because a 30-60-90 triangle is exactly half of an equilateral triangle. If the equilateral triangle has side 2x, the altitude cuts the base into two pieces of length x, which is the short leg, while the hypotenuse is still a full side, 2x. The Pythagorean theorem then gives the long leg, x√3.
How do you find the short leg from the long leg?
Divide the long leg by √3, then rationalize if your teacher wants simplest radical form. For a long leg of 12: 12√3 = 12√33 = 4√3. Then double the short leg for the hypotenuse: 8√3. Check: the long leg should be about 1.73 times the short leg.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSG-SRT.C.6, CCSS 8.G.B.7
Cite this entry
Effortless Math Team. “30-60-90 Triangle.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/30-60-90-triangle/
Flashcard
Term30-60-90 Trianglesides 1 : √3 : 2
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