45-45-90 Triangle Calculator
Solve any 45-45-90 isosceles right triangle from a single known side using the 1 : 1 : √2 ratio.
45-45-90 Triangle Calculator
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Formula: If each leg = x, then hypotenuse = x√2
Worked example — Hypotenuse ≈ 19.799 m
Formula
If each leg = x, then hypotenuse = x√2
A 45-45-90 triangle is exactly half of a square cut along its diagonal, which permanently locks the two legs to the same length and makes the hypotenuse equal to the leg multiplied by √2. Both 45° angles are identical, so the triangle is isosceles as well as right-angled. Because the hypotenuse is the diagonal of the underlying square, this triangle appears whenever a square corner or grid layout needs to be measured diagonally.
Worked Examples
Example 1: Given a leg
Problem:A square corner forms a 45-45-90 triangle with a leg of 14 m. Find the hypotenuse.
Solution:Leg = 14 Hypotenuse = 14√2 ≈ 19.799
Result:Hypotenuse ≈ 19.799 m
Example 2: Given the hypotenuse
Problem:A diagonal segment in a right isosceles triangle is 18 in. Find each leg.
Solution:Leg = 18 ÷ √2 ≈ 12.7279 Both legs are equal in a 45-45-90 triangle.
Result:Each leg ≈ 12.7279 in
Frequently Asked Questions
What is a 45-45-90 triangle?
A 45-45-90 triangle is an isosceles right triangle. Its two legs are equal, and the hypotenuse is always the leg multiplied by √2.
What is the side ratio for a 45-45-90 triangle?
If each leg is x, then the hypotenuse is x√2. The fixed side ratio is 1 : 1 : √2.
How do you find the hypotenuse from a leg?
Multiply the leg by √2. If a leg is 9, the hypotenuse is 9√2 ≈ 12.7279.
Why is a leg usually written as h√2/2 instead of h/√2?
The two expressions are equal — multiplying h/√2 by √2/√2 clears the irrational denominator and gives h√2/2, the rationalised form textbooks prefer. Either way a hypotenuse of 20 yields legs of about 14.1421. The rationalised version is easier to evaluate by hand and keeps rounding error out of the denominator.
Where is this triangle used?
45-45-90 triangles appear in squares, diagonal measurements, drafting, layout work, and many geometry and trigonometry problems.
How is the 1 : 1 : √2 ratio derived?
Apply the Pythagorean theorem to a right triangle whose legs are both x: x² + x² = c², so c² = 2x² and c = x√2. The legs must be equal because the two base angles are both 45°, and equal angles face equal sides. Since √2 is irrational, a 45-45-90 triangle with whole-number legs never has a whole-number hypotenuse.
How do you get the area and perimeter from the hypotenuse alone?
Each leg is h/√2, so the area works out to h²/4 and the perimeter to h(√2 + 1). For a hypotenuse of 20 the legs are about 14.1421, the area is 400/4 = 100 square units and the perimeter is roughly 48.2843. Computing from h directly avoids rounding the leg first and then carrying that error into two further results.
Why is the hypotenuse the same as a square diagonal?
Splitting a square along a diagonal produces two congruent 45-45-90 triangles whose legs are the sides of the square and whose shared hypotenuse is that diagonal. So a square of side s has a diagonal of s√2 ≈ 1.4142s. Reversed, a known diagonal gives the side as d/√2, which is how a layout can be checked for true square corners.
What exact trigonometric values come out of this triangle?
Because the two legs are equal, sin 45° = cos 45° = 1/√2 = √2/2 ≈ 0.7071, and tan 45° = 1 exactly. That equality of sine and cosine is why the unit circle is symmetric about the 45° line, and it is the reason a 45° slope has a rise exactly equal to its run.
Where does this triangle show up in construction and layout work?
Anywhere a 45° cut is made or a square corner is checked diagonally. Mitred corners on frames and trim meet at 45°, so each half is a 45-45-90 triangle. Diagonal bracing across a square bay has a length of side × √2, and measuring both diagonals of a rectangle to confirm they match is the standard way to verify that corners are square.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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