Last updated: July 31, 2026
Isosceles Right Triangle Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
For leg a in a 45-45-90 triangle: hypotenuse = a√2, area = a²/2, perimeter = a(2 + √2), height to hypotenuse = a/√2
Where:
- a=Equal leg length
- c=Hypotenuse
- h=Height from right angle to hypotenuse
Worked Examples
Leg length 5
A classic 45-45-90 triangle example using a leg of 5 units.
- 1Multiply the leg by √2 to find the hypotenuse.
- 2Square the leg and divide by 2 for area.
- 3Add both legs and the hypotenuse for perimeter.
Leg length 8.5
Use the same formulas with a decimal leg length.
- 1Compute 8.5√2.
- 2Evaluate 8.5²/2.
- 3Find the altitude to the hypotenuse with 8.5/√2.
Small leg length
A leg of 1 gives the unit 45-45-90 reference triangle.
- 1Apply the same fixed-angle relationships.
- 2Notice that the hypotenuse is exactly √2 before rounding.
- 3Read the remaining outputs directly.
Invalid leg example
A non-positive leg does not form a triangle.
- 1Check that the leg is finite.
- 2Reject zero because side lengths must be positive.
- 3Return safe numeric defaults and an error string.
Introduction
An isosceles right triangle has two equal legs and one right angle, so its acute angles are always 45 degrees. With just one leg length, you can determine every major measurement of the triangle using fixed 45-45-90 relationships.
What Makes This Triangle Special
A 45-45-90 triangle is both right and isosceles, so symmetry makes the geometry especially clean.
Both legs are equal
The acute angles are 45° each
The hypotenuse is longer than each leg by a factor of √2
Core Formulas
Every output comes from the single input leg a.
Hypotenuse = a√2
Area = a²/2
Perimeter = a(2 + √2)
Height to hypotenuse = a/√2
Angle Results
The angle outputs do not depend on the leg length because all isosceles right triangles are similar.
Base angle = 45°
Vertex angle = 90°
Changing scale does not change angle measures
Height to the Hypotenuse
Dropping an altitude from the right angle to the hypotenuse creates two smaller similar triangles and gives a simple altitude formula.
h = a² / c
Since c = a√2, the height simplifies to a/√2
This is also a√2/2
The height to the hypotenuse is always shorter than either leg but longer than half a leg.
How to Use This Calculator
Enter one positive leg length and read the six outputs immediately.
Type the leg length
Run the calculation
Use the hypotenuse as the primary result
Check area, perimeter, and altitude for supporting geometry
Validation Rules
The only required condition is a positive finite leg length.
Zero is invalid
Negative lengths are invalid
Non-numeric entries return defaults with an error message
Common Uses
This triangle appears frequently in trigonometry, drafting, square diagonals, and coordinate geometry.
Checking square diagonal lengths
Classroom triangle problems
Fast right-triangle estimations
CAD and layout work
FAQs
Why are the acute angles always 45°?
Because the two legs are equal, the two acute angles must be equal, and together they sum to 90°.
How do you get the hypotenuse formula?
Apply the Pythagorean theorem: c² = a² + a² = 2a², so c = a√2.
Why is the area a²/2?
A right triangle has area 1/2 × base × height, and here both base and height are the leg length a.
What is the height to the hypotenuse?
It is the altitude drawn from the right-angle vertex to the hypotenuse, and it equals a/√2.
Can I use decimals for the leg length?
Yes. Any positive finite decimal input works, and results are rounded to 6 decimals.
What happens if the leg is zero or negative?
The calculator returns zeros and an error because non-positive lengths do not form a valid triangle.
Is this the same as a 45-45-90 triangle calculator?
Yes. Isosceles right triangle and 45-45-90 triangle refer to the same shape.