Calculate all sides, area, and perimeter of a 45-45-90 isosceles right triangle. Enter any side to find the others using exact ratios.
Last updated: April 2026 | By Patchworkr Team
Both legs are equal (isosceles), each opposite a 45° angle. The hypotenuse is opposite the 90° angle.
A 45-45-90 triangle is a special isosceles right triangle where two angles measure 45° and one angle measures 90°. Because two angles are equal, the two legs opposite those angles are also equal in length.
The sides of a 45-45-90 triangle always follow the ratio 1 : 1 : √2, where:
This triangle is the most common special right triangle, appearing in squares cut diagonally, coordinate geometry, and countless engineering applications. Its symmetry and predictable ratios make it ideal for quick mental calculations.
Determine if you have one leg or the hypotenuse (both legs are equal in 45-45-90).
Why: Since both legs are identical, you only need one value. Your starting point determines which calculation path to follow.
Know that: leg1 = leg2 = x, hypotenuse = x√2. Both legs are always equal.
Why: This isosceles right triangle has two equal sides by definition. The ratio comes from trigonometry and applies universally.
If leg known: hypotenuse = leg × √2. If hypotenuse known: leg = hypotenuse / √2 = hypotenuse × (√2 / 2)
Why: Formulas directly implement 1:1:√2 ratio. You only need one calculation since both legs are identical.
Area = leg² / 2; Perimeter = 2 × leg + hypotenuse
Why: Area represents half the square (two equal-leg right triangle). Perimeter sums all three sides.
Check: (leg)² + (leg)² = (hypotenuse)² using Pythagorean theorem; 2leg² = hyp².
Why: Verification catches errors. All right triangles satisfy Pythagorean relation.
Steel Frame Diagonal Bracing
The sides are in the ratio 1 : 1 : √2, where both legs equal x and the hypotenuse equals x√2.
Multiply the leg by √2 (approximately 1.414). For example, if the leg is 8, the hypotenuse is 8√2 ≈ 11.31.
It has two equal sides (isosceles) and one 90° angle (right triangle). The two equal angles are each 45°.
Yes! For legs of length x: x² + x² = hypotenuse², which gives hypotenuse = √(2x²) = x√2.
Diagonals of squares, baseball diamonds, corner braces in construction, and any time you cut a square in half diagonally.
Divide the hypotenuse by √2 to get each leg. For a hypotenuse of 14, each leg is 14 / √2 ≈ 9.90.
Yes! At 45° on the unit circle, both sin(45°) and cos(45°) equal √2/2, derived from this triangle's ratios.
Use Area = leg² / 2. Since both legs are equal, it's half the area of a square with the same side length.
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