How do I solve special right triangles?
Pick the 30-60-90 or the 45-45-90 triangle, say which part 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 sides 5, 8.660254 and 10 has an area of 21.650635.
- 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 sides 5, 8.660254 and 10 has an area of 21.650635.
What does the triangle look like?
How to calculate
Solves a 30-60-90 or 45-45-90 triangle from one side, the area or the perimeter: all three sides with exact square roots, the area, the perimeter and the height.
Example with the default inputs (Triangle 30-60-90, You know the Short leg a, Its value 5): A 30°, 60°, 90° triangle with sides 5, 8.660254 and 10 has an area of 21.650635.
Method: 30-60-90: a : b : c = 1 : √3 : 2, area = a²√3 ÷ 2, perimeter = a(3 + √3). 45-45-90: a : b : c = 1 : 1 : √2, area = a² ÷ 2, perimeter = a(2 + √2). Height to the hypotenuse = a × b ÷ c.
- a is the side opposite the smallest angle (30° or 45°), b the other leg, 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.
- Triangle 30-60-90, 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; 45-45-90 sides s, s, √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
- Triangle 45-45-90, You know the Leg a, Its value 5 gives Short leg a 5, Long leg b 5, Hypotenuse c 7.071068, Area 12.5, Perimeter 17.071068, Exact sides a = 5, b = 5, c = 5√2, Angles 45°, 45°, 90°.Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s; 45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry
- Triangle 30-60-90, You know the Long leg b, Its value 6 gives Short leg a 3.464102, Hypotenuse c 6.928203, Area 10.392305, 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; 45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry
- Triangle 45-45-90, You know the Area, Its value 50 gives Short leg a 10, Long leg b 10, Hypotenuse c 14.142136, Perimeter 34.142136.Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s; 45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry
- Triangle 30-60-90, 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; 45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry
How it works
Pick the triangle, say which part you know, and type its value v (from 0.000001 to 1,000,000,000). Call the legs a (opposite the smallest angle, 30° or 45°) and b, and the hypotenuse c.
30-60-90 triangle. a : b : c = 1 : √3 : 2. The short leg a 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.
45-45-90 triangle. a : b : c = 1 : 1 : √2. The leg a comes from the known part:
- leg: a = v
- hypotenuse: a = v ÷ √2
- area: a = √(2v), because the area is a² ÷ 2
- perimeter: a = v ÷ (1 + 1 + √2)
Then b = a and c = a√2.
The part you typed keeps its exact value. The other parts 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°" or "45°, 45°, 90°". Side ratio: "1 : √3 : 2" or "1 : 1 : √2".
The maths runs in floating point (√3 and √2 as 64-bit numbers); values round for display only.
Exact sides
When the known part is a side, the page also writes the three sides with square roots, as a = …, b = …, c = …. The typed value is read as its shortest decimal (5, 2.5, 0.125) and turned into an exact fraction r. Each side is a fraction times √1, √2 or √3:
| Triangle | Known | a | b | c |
|---|---|---|---|---|
| 30-60-90 | short leg r | r | r√3 | 2r |
| 30-60-90 | long leg r | (r ÷ 3)√3 | r | (2r ÷ 3)√3 |
| 30-60-90 | hypotenuse r | r ÷ 2 | (r ÷ 2)√3 | r |
| 45-45-90 | leg r | r | r | r√2 |
| 45-45-90 | hypotenuse r | (r ÷ 2)√2 | (r ÷ 2)√2 | r |
Each fraction p/q is in lowest terms. It is written p (q = 1) or p/q; times a root it is written p√k (q = 1), p√k/q, and √k or √k/q when p = 1. Examples: 7√3, 2√3, √3/3, 5√2, 5/2, 5√3/2.
Exact sides show only when the typed value has at most 6 digits after the decimal point. From the area or the perimeter there are no exact sides.
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
A 30-60-90 triangle with short leg 7 (OpenStax Example 5: the side opposite 30° is 7). b = 7√3 ≈ 12.124, c = 2 × 7 = 14. Area = 7 × 7√3 ÷ 2 ≈ 42.435. Perimeter = 7 + 7√3 + 14 ≈ 33.124. Height = 7 × 7√3 ÷ 14 ≈ 6.062. Exact: a = 7, b = 7√3, c = 14.
A 45-45-90 triangle with legs 5. c = 5√2 ≈ 7.0711. Area = 25 ÷ 2 = 12.5. Perimeter = 10 + 5√2 ≈ 17.071. Exact: a = 5, b = 5, c = 5√2.
A 30-60-90 triangle with long leg 6. a = 6 ÷ √3 = 2√3 ≈ 3.4641, c = 2a = 4√3 ≈ 6.9282. Area = 2√3 × 6 ÷ 2 = 6√3 ≈ 10.392. Height = 2√3 × 6 ÷ 4√3 = 3.
A 45-45-90 triangle with area 50. a² ÷ 2 = 50, so a = b = 10 and c = 10√2 ≈ 14.142; perimeter = 20 + 10√2 ≈ 34.142.
A 30-60-90 triangle with perimeter 30. a = 30 ÷ (3 + √3) ≈ 6.3397, b = a√3 ≈ 10.981, c = 2a ≈ 12.679, area = a × b ÷ 2 ≈ 34.808.
Other questions people ask
What are the special right triangles?
The 30-60-90 triangle and the 45-45-90 triangle. Their sides are always in the same ratio: 1 : √3 : 2 for the 30-60-90 triangle and 1 : 1 : √2 for the 45-45-90 triangle, so one side gives the other two.
Where do the ratios come from?
A 30-60-90 triangle is half of an equilateral triangle, so the hypotenuse is twice the short leg, and the Pythagorean theorem gives the long leg: √(2² − 1²) = √3. A 45-45-90 triangle is half of a square, so its legs are equal and the hypotenuse is √(1² + 1²) = √2.
How do I find the sides of a 30-60-90 triangle 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 a short leg of 5 and a long leg of 5√3 ≈ 8.66.
How do I find the hypotenuse of a 45-45-90 triangle?
Multiply a leg by √2. Legs of 5 give a hypotenuse of 5√2 ≈ 7.071. To go back, divide the hypotenuse by √2: 10 ÷ √2 = 5√2 ≈ 7.071.
Why is the long leg divided by √3 written as a multiple of √3?
Dividing by √3 is the same as multiplying by √3 ÷ 3, which removes the root from the bottom. A long leg of 6 gives a short leg of 6 ÷ √3 = 2√3.
Can I start from the area or the perimeter?
Yes. For a 45-45-90 triangle the area is a² ÷ 2, so an area of 50 gives legs of 10. For a 30-60-90 triangle the perimeter is a(3 + √3), so a perimeter of 30 gives a short leg of 30 ÷ (3 + √3) ≈ 6.34.