How do I solve a 45-45-90 triangle?
Say which part of the 45-45-90 triangle you know and type its value. The calculator gives the legs and the hypotenuse, with exact square roots, plus the area, the perimeter and the height.
- Hypotenuse c
- 7.071068
A 45-45-90 triangle with legs 5 has a hypotenuse of 7.071068 and an area of 12.5.
- Leg a
- 5
- Leg b
- 5
- Area
- 12.5
- Perimeter
- 17.071068
- Height to the hypotenuse
- 3.535534
- Exact sides
- a = 5, b = 5, c = 5√2
- Angles
- 45°, 45°, 90°
- Side ratio a : b : c
- 1 : 1 : √2
Hypotenuse c: 7.071068. A 45-45-90 triangle with legs 5 has a hypotenuse of 7.071068 and an area of 12.5.
What does the triangle look like?
How to calculate
Solves a 45-45-90 (isosceles right) triangle from a leg, the hypotenuse, the area or the perimeter: the sides with exact square roots, the area, the perimeter and the height.
Example with the default inputs (You know the Leg a, Its value 5): A 45-45-90 triangle with legs 5 has a hypotenuse of 7.071068 and an area of 12.5.
Method: a : b : c = 1 : 1 : √2. Area = a² ÷ 2, perimeter = a(2 + √2), height to the hypotenuse = c ÷ 2.
- a and b are the equal legs, 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 Leg a, Its value 5 gives Leg a 5, Leg b 5, Hypotenuse c 7.071068, Area 12.5, Perimeter 17.071068, Height to the hypotenuse 3.535534, Exact sides a = 5, b = 5, c = 5√2.Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry
- You know the Hypotenuse c, Its value 10 gives Leg a 7.071068, Leg b 7.071068, Area 25, Perimeter 24.142136, Exact sides a = 5√2, b = 5√2, c = 10.Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry
- You know the Area, Its value 50 gives Leg a 10, Hypotenuse c 14.142136, Perimeter 34.142136, Height to the hypotenuse 7.071068.Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry
- You know the Leg a, Its value 1 gives Hypotenuse c 1.414214, Area 0.5, Angles 45°, 45°, 90°, Side ratio a : b : c 1 : 1 : √2.Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry: the 45-45-90 triangle with legs 1 has hypotenuse √2
How it works
Say which part you know and type its value v (from 0.000001 to 1,000,000,000). The legs a and b are equal and meet at the 90° angle; c is the hypotenuse; a : b : c = 1 : 1 : √2. The leg 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 others follow from a:
- Area = a × b ÷ 2 (in the unit squared).
- Perimeter = a + b + c.
- Height to the hypotenuse = a × b ÷ c.
- Angles: "45°, 45°, 90°". Side ratio: "1 : 1 : √2".
The maths runs in floating point (√2 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 |
|---|---|---|---|
| leg r | r | r | r√2 |
| 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 √2 it is written p√2 (q = 1), p√2/q, and √2 or √2/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
Legs 5. c = 5√2 ≈ 7.0711. Area = 25 ÷ 2 = 12.5. Perimeter = 10 + 5√2 ≈ 17.071. Height = 25 ÷ 5√2 ≈ 3.5355. Exact: a = 5, b = 5, c = 5√2.
Hypotenuse 10. a = b = 10 ÷ √2 = 5√2 ≈ 7.0711. Area = 50 ÷ 2 = 25. Perimeter = 10 + 10√2 ≈ 24.142. Exact: a = 5√2, b = 5√2, c = 10.
Area 50. a = √100 = 10, c = 10√2 ≈ 14.142, perimeter = 20 + 10√2 ≈ 34.142, height = 10 ÷ √2 ≈ 7.0711.
Legs 1. c = √2 ≈ 1.4142, area = 0.5.
Other questions people ask
What is the 45-45-90 triangle rule?
The two legs are equal and the hypotenuse is a leg times √2, so the sides are in the ratio 1 : 1 : √2. It is half of a square, cut along the diagonal.
How do I find the hypotenuse from a leg?
Multiply the leg by √2. Legs of 5 give a hypotenuse of 5√2 ≈ 7.071.
How do I find the legs from the hypotenuse?
Divide the hypotenuse by √2, which is the same as multiplying by √2 ÷ 2. A hypotenuse of 10 gives legs of 5√2 ≈ 7.071.
What is the area of a 45-45-90 triangle?
Half the square of a leg: a² ÷ 2. Legs of 5 give 12.5. From the area back, a = √(2 × area), so an area of 50 gives legs of 10.
Is a 45-45-90 triangle isosceles?
Yes. It is the isosceles right triangle: two equal legs meet at the 90° angle, and the two other angles are 45° each.
What is the height to the hypotenuse?
Half the hypotenuse: a × a ÷ (a√2) = a ÷ √2 = c ÷ 2. Legs of 1 give a hypotenuse of √2 ≈ 1.414 and a height of about 0.7071.