How can I tell which angle is 30 degrees and which is 60 degrees in a real-world triangle?

The 30 degree angle is opposite the shortest side, and the 60 degree angle is opposite the longer leg. The right angle (90 degrees) is always opposite the hypotenuse (the longest side).

Why does the long leg equal short leg x sqrt(3)?

The 30-60-90 triangle is half of an equilateral triangle split along its height. The height of an equilateral triangle with side 2s is s x sqrt(3), which becomes the long leg.

What if I know the hypotenuse but not the short leg?

Divide the hypotenuse by 2 to get the short leg, then multiply the short leg by sqrt(3) to get the long leg.

30-60-90 Triangle Calculator

Solve a 30-60-90 right triangle from the shortest leg. Returns the long leg and hypotenuse using the special ratio 1:sqrt(3):2. Essential for geometry, trigonometry, and construction layouts.

By AbraCalc Editorial Team · Reviewed by AbraCalc Methodology Review · Last reviewed: 2026-06-25

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How to use this tool

Enter short leg (opposite 30 deg) in the fields above.
Results update instantly as you type — or click Calculate.
Read your hypotenuse (opposite 90 deg) and the full breakdown beneath it.

Formula

long leg = short leg * sqrt(3); hypotenuse = short leg * 2

How it works

Ratio 1:√3:2. Given short leg a: long leg b = a√3, hypotenuse c = 2a.

Worked example

Short leg = 5

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o
n
g
l
e
g
=
5
*
s
q
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t
(
3
)
=
8
.
6
6
0
.
H
y
p
o
t
e
n
u
s
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=
1
0
.

Common mistakes to avoid

Confusing which side is the short leg versus the long leg — the side opposite the 30 degree angle is the short leg, and the side opposite the 60 degree angle is the long leg.
Using the long leg as input and multiplying by sqrt(3) again instead of dividing, producing a result that is 3 times too large.
Mixing up the hypotenuse formula: the hypotenuse is 2 x short leg, not 2 x long leg.

Key terms

Frequently asked questions

How can I tell which angle is 30 degrees and which is 60 degrees in a real-world triangle?: The 30 degree angle is opposite the shortest side, and the 60 degree angle is opposite the longer leg. The right angle (90 degrees) is always opposite the hypotenuse (the longest side).
Why does the long leg equal short leg x sqrt(3)?: The 30-60-90 triangle is half of an equilateral triangle split along its height. The height of an equilateral triangle with side 2s is s x sqrt(3), which becomes the long leg.
What if I know the hypotenuse but not the short leg?: Divide the hypotenuse by 2 to get the short leg, then multiply the short leg by sqrt(3) to get the long leg.