30-60-90 Triangle Calculator
Enter any one side of a 30-60-90 triangle to find all three sides and the area. This special right triangle has sides in the ratio 1 : √3 : 2.
Understanding the 30-60-90 Triangle
The 30-60-90 triangle is one of two special right triangles (the other being 45-45-90). Its angles are 30 degrees, 60 degrees, and 90 degrees, and its sides follow the exact ratio 1 : √3 : 2. This ratio holds for all 30-60-90 triangles regardless of size.
The short leg is opposite the 30-degree angle, the long leg is opposite the 60-degree angle, and the hypotenuse is opposite the 90-degree angle. Once you know any one side, you can find the others by multiplying by the appropriate factor from the ratio.
This triangle arises naturally when you bisect an equilateral triangle. Draw an equilateral triangle with side 2, then drop a perpendicular from one vertex to the midpoint of the opposite side. That perpendicular has length √3, and the two halves are 30-60-90 triangles with sides 1, √3, and 2.
Practical Applications of 30-60-90 Triangles
In architecture, roof pitches and staircases often use 30-60-90 angles for structural and aesthetic reasons. The predictable ratios simplify construction and material estimation. Designers know that a 30-degree slope requires specific rise-to-run ratios.
Hexagonal patterns in tilework and engineering rely on 30-60-90 triangles. A regular hexagon can be divided into six equilateral triangles, each of which splits into two 30-60-90 triangles. This makes hexagonal dimensions easy to calculate.
In trigonometry, the 30-60-90 triangle provides exact values for sine, cosine, and tangent at 30 and 60 degrees. These values (like sin 30° = ½, cos 60° = ½, tan 60° = √3) are derived directly from the side ratios.
30-60-90 vs. 45-45-90 Triangles
The 45-45-90 triangle has two 45-degree angles and a right angle, with sides in the ratio 1 : 1 : √2. It is isosceles, meaning the two legs are equal. The 30-60-90 triangle has unequal legs and a different angle configuration.
Both are special because all side ratios are known exactly without trigonometry. For 45-45-90, if the leg is s, the hypotenuse is s√2. For 30-60-90, if the short leg is s, the long leg is s√3 and the hypotenuse is 2s.
Memorizing both ratios speeds up geometry and trigonometry problems. This calculator handles 30-60-90 specifically. If you need 45-45-90 calculations, use an isosceles right triangle calculator or a general right triangle tool with equal legs.
Frequently Asked Questions
What is a 30-60-90 triangle?
A 30-60-90 triangle is a right triangle with angles of 30 degrees, 60 degrees, and 90 degrees. The sides are in the ratio 1 : √3 : 2.
How do you find the sides of a 30-60-90 triangle?
If the short leg (opposite 30°) is s, the long leg (opposite 60°) is s√3, and the hypotenuse is 2s. Multiply the short leg by √3 to get the long leg and by 2 to get the hypotenuse.
Where does the 1 : √3 : 2 ratio come from?
It comes from bisecting an equilateral triangle. The altitude of an equilateral triangle with side 2 is √3, creating two 30-60-90 triangles with sides 1, √3, and 2.
Can I use the Pythagorean theorem to verify the ratio?
Yes. For sides 1, √3, 2: 1² + (√3)² = 1 + 3 = 4 = 2². The Pythagorean theorem confirms the relationship.
What is the area formula for a 30-60-90 triangle?
Area = ½ × short leg × long leg = ½ × s × s√3 = (s²√3) / 2. Alternatively, use the general right triangle formula with the two legs.