How do I solve a 30-60-90 triangle?
Say which part of the 30-60-90 triangle you know and type its value. The calculator gives all three sides, with exact square roots, plus the area, the perimeter and the height.
- Hypotenuse c
- 10
A 30-60-90 triangle with short leg 5 has a long leg of 8.660254 and a hypotenuse of 10.
- Short leg a
- 5
- Long leg b
- 8.660254
- Area
- 21.650635
- Perimeter
- 23.660254
- Height to the hypotenuse
- 4.330127
- Exact sides
- a = 5, b = 5√3, c = 10
- Angles
- 30°, 60°, 90°
- Side ratio a : b : c
- 1 : √3 : 2
Hypotenuse c: 10. A 30-60-90 triangle with short leg 5 has a long leg of 8.660254 and a hypotenuse of 10.
What does the triangle look like?
How to calculate
Solves a 30-60-90 triangle from the short leg, the long leg, the hypotenuse, the area or the perimeter: all three sides with exact square roots, the area, the perimeter and the height.
Example with the default inputs (You know the Short leg a, Its value 5): A 30-60-90 triangle with short leg 5 has a long leg of 8.660254 and a hypotenuse of 10.
Method: a : b : c = 1 : √3 : 2. Area = a²√3 ÷ 2, perimeter = a(3 + √3), height to the hypotenuse = a√3 ÷ 2.
- a is the short leg (opposite 30°), b the long leg (opposite 60°), c the hypotenuse.
- The known value is from 0.000001 to 1,000,000,000, in any one unit; the area is in that unit squared.
- Exact sides show when a side is known as a decimal with at most 6 places.
Worked examples
Each example is checked against the calculator on every build.
- You know the Short leg a, Its value 7 gives Short leg a 7, Long leg b 12.124356, Hypotenuse c 14, Area 42.435245, Perimeter 33.124356, Height to the hypotenuse 6.062178, Exact sides a = 7, b = 7√3, c = 14.Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry, Example 5: the side opposite 30° is 7, so the hypotenuse is 14 and the adjacent side 7√3 ≈ 12.1
- You know the Hypotenuse c, Its value 10 gives Short leg a 5, Long leg b 8.660254, Area 21.650635, Perimeter 23.660254, Exact sides a = 5, b = 5√3, c = 10, Side ratio a : b : c 1 : √3 : 2.Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry
- You know the Long leg b, Its value 6 gives Short leg a 3.464102, Hypotenuse c 6.928203, Height to the hypotenuse 3, Exact sides a = 2√3, b = 6, c = 4√3.Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry
- You know the Perimeter, Its value 30 gives Short leg a 6.339746, Long leg b 10.980762, Hypotenuse c 12.679492, Area 34.807621.Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry
How it works
Say which part you know and type its value v (from 0.000001 to 1,000,000,000). The short leg a is opposite 30°, the long leg b opposite 60°, and the hypotenuse c opposite 90°; a : b : c = 1 : √3 : 2. The short leg comes from the known part:
- short leg: a = v
- long leg: a = v ÷ √3
- hypotenuse: a = v ÷ 2
- area: a = √(2v ÷ √3), because the area is a × a√3 ÷ 2
- perimeter: a = v ÷ (1 + √3 + 2)
Then b = a√3 and c = 2a. The part you typed keeps its exact value; the others follow from a:
- Area = a × b ÷ 2 (in the unit squared).
- Perimeter = a + b + c.
- Height to the hypotenuse = a × b ÷ c.
- Angles: "30°, 60°, 90°". Side ratio: "1 : √3 : 2".
The maths runs in floating point (√3 as a 64-bit number); values round for display only.
Exact sides
When the known part is a side, the page also writes the sides with square roots, as a = …, b = …, c = …. The typed value is read as its shortest decimal and turned into an exact fraction r:
| Known | a | b | c |
|---|---|---|---|
| short leg r | r | r√3 | 2r |
| long leg r | (r ÷ 3)√3 | r | (2r ÷ 3)√3 |
| hypotenuse r | r ÷ 2 | (r ÷ 2)√3 | r |
Each fraction p/q is in lowest terms. It is written p (q = 1) or p/q; times √3 it is written p√3 (q = 1), p√3/q, and √3 or √3/q when p = 1. Exact sides show only when the typed value has at most 6 digits after the decimal point, and not from the area or the perimeter.
Rules
- The value is from 0.000001 to 1,000,000,000. A value outside that range shows a message on its field.
Worked examples by hand
Short leg 7 (OpenStax Example 5). b = 7√3 ≈ 12.124, c = 14. Area = 49√3 ÷ 2 ≈ 42.435. Perimeter = 21 + 7√3 ≈ 33.124. Height = 7 × 7√3 ÷ 14 ≈ 6.062. Exact: a = 7, b = 7√3, c = 14.
Hypotenuse 10. a = 5, b = 5√3 ≈ 8.6603. Area = 25√3 ÷ 2 ≈ 21.651. Perimeter = 15 + 5√3 ≈ 23.660. Exact: a = 5, b = 5√3, c = 10.
Long leg 6. a = 6 ÷ √3 = 2√3 ≈ 3.4641, c = 4√3 ≈ 6.9282. Height = 2√3 × 6 ÷ 4√3 = 3. Exact: a = 2√3, b = 6, c = 4√3.
Perimeter 30. a = 30 ÷ (3 + √3) ≈ 6.3397, b ≈ 10.981, c ≈ 12.679, area ≈ 34.808.
Other questions people ask
What is the 30-60-90 triangle rule?
The sides are always in the ratio 1 : √3 : 2. The short leg is opposite 30°, the long leg (short leg × √3) is opposite 60°, and the hypotenuse (short leg × 2) is opposite 90°.
How do I find the long leg from the short leg?
Multiply by √3. A short leg of 7 gives a long leg of 7√3 ≈ 12.12 and a hypotenuse of 14.
How do I find the short leg from the hypotenuse?
Halve it. A hypotenuse of 10 gives a short leg of 5 and a long leg of 5√3 ≈ 8.66.
How do I find the short leg from the long leg?
Divide by √3, which is the same as multiplying by √3 ÷ 3. A long leg of 6 gives a short leg of 2√3 ≈ 3.464 and a hypotenuse of 4√3 ≈ 6.928.
Why are the sides 1, √3 and 2?
A 30-60-90 triangle is half of an equilateral triangle with sides 2. Cutting it in half gives a short leg of 1 and a hypotenuse of 2, and the Pythagorean theorem gives the third side: √(4 − 1) = √3.
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 7 that is 49√3 ÷ 2 ≈ 42.44.