Why is the hypotenuse exactly leg x sqrt(2)?

By the Pythagorean theorem, hyp squared = leg squared + leg squared = 2 x leg squared, so hyp = leg x sqrt(2). This is the defining property of all 45-45-90 triangles.

If I know the hypotenuse, how do I find the leg?

Divide the hypotenuse by sqrt(2), or equivalently multiply it by sqrt(2)/2 which is approximately 0.7071. Both legs are equal.

Where do 45-45-90 triangles appear in carpentry?

Any time you cut a square piece diagonally — such as mitering a picture frame corner or laying diagonal floor tiles — you create a 45-45-90 triangle. The diagonal cut length is the leg times sqrt(2).

45-45-90 Triangle Calculator

Solve a 45-45-90 isosceles right triangle from one leg. Computes both legs and the hypotenuse using the special ratio 1:1:sqrt(2). Ideal for geometry, carpentry, and tiling problems.

By AbraCalc Editorial Team · Reviewed by AbraCalc Methodology Review · Last reviewed: 2026-06-25

Embed this tool on your site

<iframe src="https://abracalc.com/calculator/45-45-90-triangle-calculator/embed.html" width="340" height="320" style="border:1px solid #e4e7ec;border-radius:12px" loading="lazy"></iframe>

How to use this tool

Enter leg (shorter sides) in the fields above.
Results update instantly as you type — or click Calculate.
Read your hypotenuse and the full breakdown beneath it.

Formula

hypotenuse = leg * sqrt(2); both legs are equal

How it works

In a 45-45-90 triangle the two legs are equal and the hypotenuse is leg × √2.

Worked example

Leg = 5

H
y
p
o
t
e
n
u
s
e
=
5
*
s
q
r
t
(
2
)
=
7
.
0
7
1
.
B
o
t
h
l
e
g
s
=
5
.

Common mistakes to avoid

Multiplying the hypotenuse by sqrt(2) to find the leg — it is the leg that is multiplied by sqrt(2) to get the hypotenuse, not the other way around.
Entering the hypotenuse as the input when the tool asks for a leg, producing a hypotenuse that is sqrt(2) times too large.
Rounding sqrt(2) to 1.4 early in the calculation, which introduces noticeable error in precise carpentry or tiling work.

Key terms

Frequently asked questions

Why is the hypotenuse exactly leg x sqrt(2)?: By the Pythagorean theorem, hyp squared = leg squared + leg squared = 2 x leg squared, so hyp = leg x sqrt(2). This is the defining property of all 45-45-90 triangles.
If I know the hypotenuse, how do I find the leg?: Divide the hypotenuse by sqrt(2), or equivalently multiply it by sqrt(2)/2 which is approximately 0.7071. Both legs are equal.
Where do 45-45-90 triangles appear in carpentry?: Any time you cut a square piece diagonally — such as mitering a picture frame corner or laying diagonal floor tiles — you create a 45-45-90 triangle. The diagonal cut length is the leg times sqrt(2).