How to use the 30 60 90 triangle calculator
- 1 Choose which side you know: short leg, long leg or hypotenuse.
- 2 Enter its length.
- 3 Read the other two sides in the 1 : √3 : 2 ratio.
- 4 Check the area, perimeter and height.
Worked example
A 30-60-90 triangle with a short leg of 5. The long leg is 5√3 = 8.660 and the hypotenuse is 2 × 5 = 10. The area is 5 × 8.660 ÷ 2 = 21.65 and the perimeter is 23.66.
Using the 1 : √3 : 2 rule
- Find the short leg first: everything follows from it.
- Hypotenuse = 2 × short leg.
- Long leg = short leg × √3, about 1.732.
- Check with a² + b² = c².
Find circles with the Circumference Calculator, roof angles with the Roofing Calculator, and convert angles with the Angle Converter. Check side lengths with fractions in the Mixed Number Calculator.
How it works: Everything in this triangle follows from the short leg. If you enter the long leg, it is divided by the square root of 3; if you enter the hypotenuse, it is halved. The other sides, the area, the perimeter and the height to the hypotenuse are then worked out from that short leg.
Features
From any side
Short leg, long leg or hypotenuse.
1 : √3 : 2 ratio
The special right triangle rule applied for you.
Area and perimeter
Plus the height to the hypotenuse.
When to use it
Geometry homework
Check special right triangle problems. Solve any right triangle with the Pythagorean Theorem Calculator.
Trigonometry
See where sin 30° = 1/2 and cos 30° = √3/2 come from.
Carpentry
Cut 30° and 60° angles for braces and rafters. Work out a roof pitch with the Roofing Calculator.
Equilateral triangles
Half an equilateral triangle is a 30-60-90 triangle.
Related Math tasks
Frequently asked questions
What is a 30-60-90 triangle?
A right triangle with angles of 30°, 60° and 90°. Its sides are always in the ratio 1 : √3 : 2.
What are the side rules of a 30-60-90 triangle?
If the short leg is a, the long leg is a√3 and the hypotenuse is 2a.
How do I find the sides from the hypotenuse?
Halve the hypotenuse to get the short leg, then multiply the short leg by √3 for the long leg. A hypotenuse of 10 gives 5 and 8.660.
How do I find the sides from the long leg?
Divide the long leg by √3 for the short leg, then double the short leg for the hypotenuse.
Which side is opposite the 30° angle?
The short leg. The long leg is opposite 60° and the hypotenuse is opposite the right angle.
What is the area of a 30-60-90 triangle?
Half the product of the legs: (a × a√3) ÷ 2 = a²√3 ÷ 2. With a short leg of 5, the area is 21.65.
How is it related to an equilateral triangle?
Cutting an equilateral triangle in half from a corner to the middle of the opposite side makes two 30-60-90 triangles.
What is a 45-45-90 triangle?
The other special right triangle: two equal legs and a hypotenuse of leg × √2.
Why is √3 about 1.732?
√3 is irrational: 1.7320508…. The calculator uses its full value.
Can I check the answer with the Pythagorean theorem?
Yes: a² + (a√3)² = a² + 3a² = 4a² = (2a)².
What are sin, cos and tan of 30°?
sin 30° = 1/2, cos 30° = √3/2 ≈ 0.866 and tan 30° = 1/√3 ≈ 0.577, straight from the side ratios.
What are sin, cos and tan of 60°?
sin 60° = √3/2, cos 60° = 1/2 and tan 60° = √3 ≈ 1.732.
How do I find the height of an equilateral triangle?
It is the long leg of the 30-60-90 triangle formed by cutting it in half: side × √3 ÷ 2.
Can the short leg be a decimal?
Yes. Enter any positive length; the other sides follow the same ratio.
Where are 30-60-90 triangles used?
In trigonometry, drafting, carpentry (30° and 60° cuts), hexagon layouts and physics problems with ramps.
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