Gleichschenkliges-rechtwinkliges-Dreieck-Rechner

A right isosceles triangle — also called a 45-45-90 triangle — is one of two "special" right triangles every geometry student learns to recognize on sight. Its angle measures are fixed at 45°, 45°, and 90°, and its side lengths follow a fixed ratio: 1 : 1 : √2. This tutorial explains why those numbers are forced, derives the ratio from the Pythagorean theorem, and walks through the calculator's two-direction solving (leg→hypotenuse or hypotenuse→leg).

Why the angles must be 45-45-90

A right triangle has one 90° angle. The other two angles must sum to 90° (because all three add to 180°). If the triangle is also isosceles, two of its sides are equal — and in a right triangle, the two legs are the only candidate pair (the hypotenuse is always the longest side, so cannot equal either leg). Equal legs force the angles opposite those legs to be equal (base angles of an isosceles triangle are equal). Those two equal angles each must therefore be 45° to sum to 90°.

Conclusion: any right triangle with two equal legs has angles 45°, 45°, 90°, and conversely any triangle with angles 45-45-90 is right isosceles. The two conditions are equivalent.

Deriving the 1 : 1 : √2 ratio

Suppose both legs have length 1. Apply the Pythagorean theorem:

hyp² = 1² + 1² = 2, so hyp = √2 ≈ 1.4142

Scaling: if both legs have length L, then hyp = L × √2. The ratio leg : leg : hypotenuse = 1 : 1 : √2 holds for every 45-45-90 triangle, regardless of scale.

Two directions of solving

This calculator accepts either the leg or the hypotenuse — not both. From whichever one you enter, the other and every derived value are computed:

• From leg L: hypotenuse = L√2, area = L²/2, perimeter = 2L + L√2.
• From hypotenuse H: leg = H/√2 = H√2/2, area = H²/4, perimeter = H√2 + H.

Entering both leg and hypotenuse produces an error if they are inconsistent (e.g. leg = 5, hyp = 6 — but 5√2 ≈ 7.07, not 6).

Worked examples

Example 1 — From a leg: L = 5. Hypotenuse = 5√2 ≈ 7.0711. Area = 5²/2 = 12.5. Perimeter = 10 + 5√2 ≈ 17.0711.

Example 2 — From a hypotenuse: H = 10. Leg = 10/√2 = 10√2/2 = 5√2 ≈ 7.0711. Area = 100/4 = 25. Perimeter = 10 + 10√2 ≈ 24.1421.

Example 3 — Inverse check: if you plug Example 1's hypotenuse (5√2) back as input, you should recover leg = 5. This is a quick algebra round-trip.

Where you actually see this triangle

• The diagonal of a unit square. A square with side 1 has a diagonal of √2 — derived by splitting the square along that diagonal into two 45-45-90 triangles.
• Carpenter's and drafter's squares. The classic 45° triangle set square used in drafting is exactly a 45-45-90.
• Tile patterns. Square tiles cut along the diagonal produce two 45-45-90 triangles, the basis of many decorative patterns and tessellations.
• Paper folding (origami). A single diagonal fold on a square sheet creates two 45-45-90 triangles. Most basic origami crease patterns rely on this.
• Trigonometry of 45°. sin 45° = cos 45° = √2/2 and tan 45° = 1. These exact values come directly from the 1 : 1 : √2 ratio.

45-45-90 vs 30-60-90

The two famous special right triangles you memorize in geometry:

Both are constructed by cutting an equilateral or isosceles right configuration. Both let you skip the calculator if the problem gives you "round" numbers.

Common mistakes

• Mistaking √2 for 2. The hypotenuse is leg × √2 ≈ 1.414 × leg, NOT 2 × leg. The diagonal of a unit square is about 1.41, not 2.
• Using leg = hyp × √2 (backwards). Dividing — leg = hyp / √2 — is the correct inversion. Rationalize: leg = hyp × √2 / 2.
• Entering both leg and hypotenuse. Pick one. The calculator computes the other.
• Assuming any right triangle with an angle close to 45° is 45-45-90. Both non-right angles must be exactly 45°. A triangle with angles 44-46-90 is right but not isosceles.