45-45-90 Triangle Calculator

Introduction

A 45-45-90 triangle (also called an isosceles right triangle) is a special right triangle with two equal legs and angles of 45°, 45°, and 90°. The side ratio is always 1 : 1 : √2, meaning if each leg is s, the hypotenuse is s√2. This fixed relationship makes it one of the most important patterns in geometry, appearing in square diagonals, coordinate rotations, unit circle analysis, and countless construction and engineering problems.

The 1 : 1 : √2 Ratio

Every 45-45-90 triangle has sides in the ratio leg : leg : hypotenuse = 1 : 1 : √2. Because both legs are equal, this is an isosceles right triangle — the simplest special right triangle.

• If each leg = a, then hypotenuse = a√2 ≈ a×1.414214

• Example: Legs = 5 each → hypotenuse = 5√2 ≈ 7.071068

• Working backwards: If hypotenuse = h, then each leg = h/√2 = h√2/2

• Example: Hypotenuse = 10 → each leg = 10/√2 ≈ 7.071068

• The ratio is constant regardless of triangle size — scaling preserves the 1:1:√2 proportion

• This makes 45-45-90 the easiest special triangle to recognize and solve

Square Diagonal Connection

The 45-45-90 triangle arises when you draw a diagonal across a square. This is why square diagonal problems and 45-45-90 triangles are inseparable.

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  Start with a square of side length s

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  Draw a diagonal from one corner to the opposite corner

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  The diagonal cuts the square into two congruent 45-45-90 triangles

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  Each triangle has two legs of length s (the square's sides) and hypotenuse = diagonal

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  By Pythagorean theorem: diagonal² = s² + s² = 2s², so diagonal = s√2

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  This proves the 1:1:√2 ratio: side:side:diagonal = s:s:s√2 = 1:1:√2

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  Any isosceles right triangle can be thought of as half a square

How This Calculator Solves It

Select whether you know a leg or the hypotenuse, enter the value, and the calculator applies the ratio to find all dimensions with 6-decimal precision.

**From leg a**:

Other leg = a (equal), Hypotenuse = a√2, Area = a²/2 (half the square's area)

**From hypotenuse h**:

Each leg = h/√2 = h√2/2, Area = h²/4

**Perimeter**:

Always = 2a + a√2 = a(2 + √2) ≈ a×3.414214

Example:

If leg = 8, calculator returns hypotenuse ≈ 11.313708, area = 32, perimeter ≈ 27.313708

Exact vs. Decimal Form

Like all special triangles, 45-45-90 benefits from keeping exact radical form in symbolic mathematics while using decimals for practical measurements.

**Exact**:

Hypotenuse = 5√2 (preferred in exams and proofs)

**Decimal**:

Hypotenuse ≈ 7.071068 (needed for construction, CAD)

√2 is irrational (≈ 1.414214) — cannot be expressed as a fraction

Rounding errors:

(√2)² = exactly 2, but (1.414)² = 1.999396

Rationalize denominators:

Write h√2/2 instead of h/√2 in formal math

Area = a²/2 is exact — no radicals needed for area calculation

Real-World Applications

The 45-45-90 triangle appears wherever square diagonals, equal-component motion, or 45° angles are involved.

**Architecture**:

Diagonal bracing in square frames — if frame is 10×10 ft, diagonal = 10√2 ≈ 14.142 ft

**Carpentry**:

Cutting square stock diagonally produces 45-45-90 triangular pieces

**Coordinate geometry**:

Moving 5 units right and 5 units up gives distance 5√2 from start

**Unit circle**:

Coordinates at 45° are (√2/2, √2/2), from a 45-45-90 triangle with hypotenuse 1

**Physics**:

Projectile at 45° has equal horizontal and vertical velocity components

**Navigation**:

45° bearing splits distance equally into north/south and east/west

**Construction**:

Staircases at 45° have rise = run (each leg equal)

45-45-90 vs. 30-60-90

Use 45-45-90 for square-based problems and 30-60-90 for equilateral-triangle-based problems. Knowing when to use each saves significant time.

Trigonometric Values from 45-45-90

All trig ratios for 45° come directly from this triangle's geometry, making it fundamental to trigonometry education.

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  sin(45°) = cos(45°) = opposite/hypotenuse = 1/√2 = √2/2 ≈ 0.707107

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  tan(45°) = opposite/adjacent = 1/1 = 1 exactly

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  These are the only angles where sine and cosine are equal

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  Used in calculus, physics, and engineering constantly

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  Unit circle at 45°: Coordinates are (√2/2, √2/2) from this triangle

Common Mistakes

Avoid these frequent errors when working with 45-45-90 triangles:

**Mistake**:

Thinking it applies to all right triangles. Fix: Only for isosceles right triangles (equal legs)

**Mistake**:

Forgetting to multiply by √2 for hypotenuse. Fix: If leg = 6, hypotenuse = 6√2, not 6 or 12

**Mistake**:

Rounding √2 to 1.4. Fix: Use 1.414214 or keep symbolic √2

**Mistake**:

Confusing with 30-60-90 (ratio 1:√3:2). Fix: 45-45-90 has equal legs, 30-60-90 doesn't

**Mistake**:

Thinking area = a². Fix: Area = a²/2 (half the square, not the full square)

**Mistake**:

Writing h/√2 instead of rationalizing to h√2/2. Fix: Formal math prefers rationalized form