The 45-45-90 triangle is one of the special right triangles, and it is the shape you get whenever a right angle is bisected, which makes it a building block for everything from roof bracing to paper folding to the geometry of a square. This calculator finds the remaining sides, the area and the angles from whichever single measurement you already have.
What Is the 45-45-90 Triangle Calculator?
A 45-45-90 triangle is a right isosceles triangle: it has one right angle and two equal 45 degree angles, so its two legs are the same length. The hypotenuse is always the leg multiplied by the square root of two, roughly 1.41421.
This calculator works in either direction. Give it a leg and it finds the hypotenuse, the area and the perimeter. Give it the hypotenuse and it divides by the square root of two to recover the legs.
Why the Square Root of Two
The Pythagorean theorem gives the answer directly. With two legs of length a, the hypotenuse satisfies c squared equals a squared plus a squared, which is 2a squared, so c equals a times the square root of two.
Working the other way, if the hypotenuse is c then each leg is c divided by the square root of two, which is the same as c multiplied by the square root of two and then divided by two.
Where This Triangle Appears
It is the triangle you get by cutting a square along its diagonal, which is why it turns up in roof layouts: a 45 degree hip or rafter creates a 45-45-90 triangle at every end of the ridge.
It is also the reference triangle for the other special right triangles. The 30-60-90 triangle has a hypotenuse twice the short leg, and both are derived from the unit circle, where the 45 degree point on the circle is the corner of a square inscribed in it.
Checking Your Work
The fastest sanity check is the ratio. If you compute the hypotenuse from a leg and the answer is not close to 1.414 times the leg, something has gone wrong. A 10 unit leg should give a hypotenuse of 14.1421 and an area of 50 square units.
Another useful check is the square on the hypotenuse. Its area is twice the area of the square on a leg, which is another way of remembering the relationship without arithmetic.
Frequently Asked Questions
What is the ratio of a 45-45-90 triangle?
The sides are in the ratio 1 to 1 to the square root of two, so the two legs are equal and the hypotenuse is about 1.414 times either leg.
How do I find the hypotenuse of a 45-45-90 triangle?
Multiply the length of a leg by the square root of two, which is 1.41421. A 10 unit leg gives a hypotenuse of 14.1421 units.
How do I find the legs from the hypotenuse?
Divide the hypotenuse by the square root of two, or multiply it by 0.70711. A 20 unit hypotenuse gives legs of 14.1421 units each.
What is the area of a 45-45-90 triangle?
It is half the leg squared, since the area of a right triangle is half the base times the height and both legs are equal. Legs of 10 units give an area of 50 square units.
How is a 45-45-90 triangle different from a 30-60-90 triangle?
Both are special right triangles with fixed side ratios. In a 45-45-90 the legs are equal and the hypotenuse is the leg times the square root of two, while in a 30-60-90 the short leg is half the hypotenuse and the long leg is three quarters of it.