45 45 90 Triangle Calculator
A 45 45 90 triangle has sides locked in the ratio 1 : 1 : √2. Enter any one side — a leg or the hypotenuse — and this calculator fills in the other two, plus area and perimeter.
Example: with Which side do you know? A leg (either equal side) · Known side length 5 → Sides leg / leg / hypotenuse: 5.00 / 5.00 / 7.07.
Computed by the calculator below using its default values. Change any input to see your own numbers.
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Check it outThe 45 45 90 triangle rules
A 45-45-90 triangle is an isosceles right triangle: two 45° angles, one 90° angle, and two equal legs. The one rule that matters is the side ratio 1 : 1 : √2, which follows straight from the Pythagorean theorem: leg² + leg² = hypotenuse², so the hypotenuse is always the leg times √2 ≈ 1.41421.
That gives two one-step formulas. Know a leg? Hypotenuse = leg × √2: a leg of 5 makes the hypotenuse 5√2 ≈ 7.07. Know the hypotenuse? Leg = hypotenuse ÷ √2: a hypotenuse of 10 gives legs of 10 ÷ 1.41421 ≈ 7.07 each. Area is leg² ÷ 2 (each leg doubles as base and height), so the 5-5-7.07 triangle covers 12.5 square units. You see this shape everywhere — it’s exactly half a square cut along its diagonal, which is why a square’s diagonal is also side × √2.
How it’s calculated
Sides follow the fixed 45-45-90 ratio 1 : 1 : √2. From a leg: hypotenuse = leg × √2; from the hypotenuse: leg = hypotenuse ÷ √2 (equivalently hypotenuse × √2 ÷ 2), with √2 = 1.41421356… Area = leg² ÷ 2, since the two legs are perpendicular and serve as base and height. Perimeter = 2 × leg + hypotenuse. Results are rounded to two decimals and work in any unit.
Results update as you type and are estimates, not professional advice — verify important decisions with a qualified professional.
Common mistakes
- Multiplying the hypotenuse by √2 to get a leg — going from hypotenuse to leg you divide by √2, not multiply.
- Using the 30-60-90 ratio (1 : √3 : 2) by mistake — the 45-45-90 ratio is 1 : 1 : √2 and its two legs are equal.
- Computing area as leg² — that is the full square; the triangle is half of it, leg² ÷ 2.
Frequently asked questions
What is the 45 45 90 triangle rule?
The sides are always in the ratio 1 : 1 : √2 — two equal legs and a hypotenuse √2 (about 1.414) times as long. The angles are fixed at 45°, 45°, and 90°.
How do I find the hypotenuse of a 45 45 90 triangle?
Multiply a leg by √2. A leg of 5 gives a hypotenuse of 5 × 1.41421 ≈ 7.07.
How do I find the legs from the hypotenuse?
Divide the hypotenuse by √2. A hypotenuse of 10 gives legs of 10 ÷ 1.41421 ≈ 7.07 each — the same as multiplying by √2 ÷ 2.
Why is the ratio 1 : 1 : sqrt 2?
Because the two legs are equal, the Pythagorean theorem gives hypotenuse² = leg² + leg² = 2 × leg², so the hypotenuse is leg × √2.
Is a 45 45 90 triangle half a square?
Yes — cutting a square along its diagonal produces two 45-45-90 triangles, and the diagonal (the hypotenuse) equals the square’s side × √2.