What this calculator does
A right angled isosceles triangle, also called a 45-45-90 triangle, is a right triangle with two equal legs and two equal 45° angles alongside the right angle. That symmetry fixes the side ratio at 1 : 1 : √2, for the two equal legs and the hypotenuse, so a single known side is enough to find the rest.
This calculate right angle triangle tool is built specifically for that 45-45-90 case: enter either leg length or the hypotenuse and it returns every side, the area and the perimeter, using the fixed √2 relationship rather than requiring the general triangle formulas.
The formula
By Pythagoras, leg² + leg² = hypotenuse², so hypotenuse = leg × √2. Given the hypotenuse instead, the leg is recovered by dividing by √2. Area follows as ½ × leg², since the two equal legs form the right angle.
| Term | Meaning |
|---|---|
| Leg | Either of the two equal sides that meet at the right angle. |
| Hypotenuse | The longest side, opposite the right angle, equal to leg × √2. |
| 45-45-90 triangle | The standard name for a right isosceles triangle, after its three angles. |
The inputs explained
| Field | What to enter |
|---|---|
| Known side | Whether the length you are entering is a leg or the hypotenuse. |
| Length of that known side | The length of that known side. |
When to use it
Cutting a mitred corner or diagonal brace
A 45° mitre or a diagonal brace across a square corner forms a right isosceles triangle, and knowing one measurement gives the exact length needed for the diagonal cut.
Finding the diagonal of a square
Splitting a square along its diagonal produces two 45-45-90 triangles, so the diagonal of a square is simply its side length multiplied by √2.
A geometry problem stating one leg or the hypotenuse
A right isosceles triangle question that gives a single side is solved directly from the fixed 1 : 1 : √2 ratio, without needing separate trigonometric steps.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Hypotenuse and area as the leg length grows
The leg taken as the known side, across a range of lengths.
| Leg length | Hypotenuse | Area |
|---|---|---|
| 1 | 1.414 | 0.5000 |
| 2 | 2.828 | 2.000 |
| 5 | 7.071 | 12.500 |
| 10 | 14.142 | 50.000 |
| 15 | 21.213 | 112.500 |
| 20 | 28.284 | 200.000 |
Leg length and area from a known hypotenuse
The hypotenuse taken as the known side, across a range of lengths.
| Hypotenuse | Leg length (each equal side) | Area |
|---|---|---|
| 1.41 | 0.9998 | 0.4998 |
| 2.83 | 2.000 | 1.999 |
| 7.07 | 4.999 | 12.496 |
| 14.14 | 9.998 | 49.985 |
| 20.00 | 14.142 | 100.000 |
| 28.28 | 19.997 | 199.940 |
Questions
What is the side ratio of an isosceles right triangle?
The two legs and the hypotenuse are in the ratio 1 : 1 : √2, since the hypotenuse equals a leg multiplied by √2, by Pythagoras applied to two equal legs.
What are the angles in a right angled isosceles triangle?
The three angles are 45°, 45° and 90°. The two 45° angles are equal because the two legs opposite them are equal, and the 90° angle is the defined right angle.
Does the existing isosceles triangle calculator cover this case?
The general isosceles triangle calculator can produce a 45-45-90 result if the base is entered as exactly the leg length times √2, but it treats the leg and base as two independent inputs rather than assuming the right-angle constraint. This calculator is built specifically for the fixed 1 : 1 : √2 case, solving from a single side.
How is this different from a 30-60-90 triangle?
A 30-60-90 triangle has three different angles and sides in the ratio 1 : √3 : 2. An isosceles right triangle has two equal 45° angles and sides in the ratio 1 : 1 : √2. They are separate special right triangles with different fixed ratios and different angle sets.
For the general case with any two equal sides and any base, see the isosceles triangle calculator. For the other common special right triangle, see the 30-60-90 triangle calculator.