Quick answer
In a 45-45-90 triangle the legs are equal and the hypotenuse is a√2. With legs of 5, the hypotenuse is 7.0711 and the area is 12.5.
Enter the leg length. The calculator returns the hypotenuse and area.
Related tools: Triangle Calculator: Area, perimeter, angles and type of a triangle from its three sides. 30-60-90 Triangle Calculator: Sides of a 30°-60°-90° triangle from the short leg. Right Triangle Calculator: Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
How to use the 45-45-90 Triangle Calculator
- Enter leg length (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Leg length | A number | > 0 |
Formula and method
Formula
legs = a; hypotenuse = a√2
IEEE-754 double precision; shown to at most 6 decimal places.
Fixed side ratios of the special right triangle.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Leg length = 5.
| Result | Value |
|---|---|
| Leg | 5 |
| Hypotenuse | 7.071068 |
| Area | 12.5 |
The first line is the headline result (leg). Below it the 45-45-90 Triangle Calculator lists hypotenuse and area, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Why √2
Both legs are a, so the hypotenuse is √(a² + a²) = a√2 = 1.4142a. The triangle is half of a square cut along its diagonal, so the diagonal of a square with side 5 is 7.07.
Uses
Set squares, roof framing and tile layouts use this triangle. Area = a²/2 = 12.5. If you know the hypotenuse, the legs are hypotenuse ÷ √2, about 0.7071 times.
How the result changes with leg length
Holding every other input at the first worked example, this table changes only leg length. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Leg length | Leg | Hypotenuse | Area |
|---|---|---|---|
| 2.5 | 2.5 | 3.535534 | 3.125 |
| 5 | 5 | 7.071068 | 12.5 |
| 10 | 10 | 14.142136 | 50 |
| 20 | 20 | 28.284271 | 200 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Leg length: entering "0" shows "Leg length must be greater than 0." beside that field and no result.
Choosing between the 45-45-90 Triangle Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Triangle Calculator](/math/triangle-calculator/) | side a, side b and side c | Area |
| [30-60-90 Triangle Calculator](/math/30-60-90-triangle-calculator/) | short leg length | Short leg |
| [Right Triangle Calculator](/math/right-triangle-calculator/) | leg a and leg b | Hypotenuse c |
| [Triangle Area Calculator](/math/triangle-area-calculator/) | base and height | Area |
| [SAS Triangle Calculator](/math/sas-triangle/) | side b, side c and included angle A | Side a |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Positive length
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for leg length to confirm this.
Frequently asked questions
What is the ratio of sides?
The sides are in the ratio 1 : 1 : √2, so a leg of 5 gives a hypotenuse of 7.07.
What is the diagonal of a 10-inch square?
A 10-inch square has a diagonal of 10√2 ≈ 14.14 inches, the hypotenuse of a 45-45-90 triangle.
Is it isosceles?
Yes, it has two equal sides (the legs) and two equal 45° angles.
What does the 45-45-90 Triangle Calculator calculate?
Sides of a 45°-45°-90° triangle from one leg. It returns Leg, Hypotenuse and Area.
What formula does the 45-45-90 Triangle Calculator use?
legs = a; hypotenuse = a√2. Fixed side ratios of the special right triangle.
What do I need to enter in the 45-45-90 Triangle Calculator?
You enter Leg length.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Positive length.
What is an example of a 45-45-90 Triangle result?
With leg length 5, the leg is 5.
Related calculators
- Triangle Calculator — Area, perimeter, angles and type of a triangle from its three sides.
- 30-60-90 Triangle Calculator — Sides of a 30°-60°-90° triangle from the short leg.
- Right Triangle Calculator — Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
- Triangle Area Calculator — Area of a triangle from base and height.
- SAS Triangle Calculator — Solve a triangle from two sides and the included angle.
- Equilateral Triangle Calculator — Area, height and perimeter of an equilateral triangle.
- Hypotenuse Calculator — Hypotenuse of a right triangle from its two legs.
- Distance Between Two Points Calculator — Straight-line distance and midpoint between two points in the plane.
- Rectangle Calculator — Area, perimeter and diagonal of a rectangle.
Sources
- OpenStax, Prealgebra 2e (Rice University): perimeter, area, volume, circumference and the Pythagorean theorem
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.