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45-45-90 Triangle

Calculate the sides, area, and perimeter of a 45-45-90 special right triangle.

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Why: Understanding 45-45-90 triangle helps you make better, data-driven decisions.

How: Enter Known Side, Value (units) to calculate results.

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SPECIAL RIGHT TRIANGLE

45-45-90 — The Isosceles Right Triangle

Half a square. Ratio 1:1:√2. The diagonal of any square creates two perfect 45-45-90 triangles.

30-60-90 →Right Triangle →

📐 Common 45-45-90 Examples — Click to Load

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45-45-90 Triangle Visualization

45-45-90 Triangle90°45°45°ABCLeg: 10.0000Leg: 10.0000Hypotenuse: 14.1421
triangle_454590.sh
CALCULATED
$ solve_454590 --known="leg" --value=10.0000
Leg
10.0000
units
Hypotenuse
14.1421
units
Area
50.0000
sq units
Perimeter
34.1421
units
Ratio: 1 : 1 : √2 | Angles: 45°, 45°, 90°
Share:
45-45-90 Triangle Calculation
Leg 10.0000 → Hypotenuse 14.1421
50.0000 sq units
P = 34.1421Ratio 1:1:√2
numbervibe.com/calculators/mathematics/triangle/45-45-90

Side Lengths

Side Proportions (1:1:√2)

Properties Radar

Step-by-Step Breakdown

RATIO
Side ratio (45-45-90)
leg : leg : hypotenuse = 1 : 1 : √2
ext{Fixed} ext{ratio} ext{for} ext{isosceles} ext{right} ext{triangle}
Angles
45°, 45°, 90°
ext{Two} ext{equal} ext{acute} ext{angles}, ext{one} ext{right} ext{angle}
Given
leg = 10.0000
ext{Both} ext{legs} ext{are} ext{equal}
Hypotenuse formula
hypotenuse = leg × √2
ext{Multiply} ext{leg} ext{by} √2
RESULT
HYPOTENUSE
14.1421 units
AREA
Area formula
A = ½ × leg × leg = leg²/2
ext{Base} ext{and} ext{height} ext{both} ext{equal} ext{leg}
AREA
50.0000 sq units
PERIMETER
Perimeter
P = 2×leg + hypotenuse = 34.1421 units
ext{Sum} ext{of} ext{all} ext{three} ext{sides}

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • The 45-45-90 triangle has a fixed 1:1:√2 side ratio (leg : leg : hypotenuse)
  • It is an isosceles right triangle — two equal legs and one 90° angle
  • The diagonal of a square divides it into two congruent 45-45-90 triangles
  • Given one side, all other sides and area can be found using the ratio — no trigonometry needed
  • sin(45°) = cos(45°) = √2/2 and tan(45°) = 1 — the simplest trigonometric values

Did You Know?

⬜A square's diagonal is exactly √2 times its side — cutting a square in half along the diagonal creates two perfect 45-45-90 trianglesSource: Wolfram MathWorld
📐Carpenters use 45-45-90 for miter joints and 45° cuts — measure equal distances from a corner; if diagonal = side × √2, the corner is squareSource: Construction
🎮In computer graphics, 45° rotations and isometric views rely on 45-45-90 geometry — pixel-perfect diagonals use the 1:√2 ratioSource: Graphics
🧭When you travel equal distances north and east, your straight-line distance = distance × √2 — a 45° bearing forms a 45-45-90 triangleSource: Navigation
🏛️The Parthenon and many classical buildings use 45° angles — the golden ratio and √2 appear throughout ancient architectureSource: Architecture
🔬√2 is irrational (cannot be written as a fraction) — one of the first irrationals discovered, causing a crisis in Pythagorean mathematicsSource: History

How 45-45-90 Triangles Work

The 45-45-90 triangle derives from a square. Draw the diagonal of a square with side 1; by the Pythagorean theorem, diagonal = √2. Each half is a 45-45-90 triangle.

Given Leg → Hypotenuse

If leg = L, then hypotenuse = L × √2. Example: leg = 10 → hypotenuse ≈ 14.14.

Given Hypotenuse → Leg

If hypotenuse = H, then leg = H ÷ √2. Example: hypotenuse = 14.14 → leg ≈ 10.

Area & Perimeter

Area = leg²/2. Perimeter = 2×leg + leg√2 = leg(2 + √2).

Expert Tips

Memorize √2 ≈ 1.414

Hypotenuse ≈ 1.41 × leg. Leg ≈ hypotenuse ÷ 1.41.

Don't Confuse with 30-60-90

30-60-90 has ratio 1:√3:2. In 45-45-90, both legs are equal.

Square Diagonal Shortcut

Diagonal of square with side s = s√2. Side = diagonal ÷ √2.

Rationalize When Needed

leg = hypotenuse/√2 = hypotenuse×√2/2.

45-45-90 vs 30-60-90 vs General Right

Property45-45-9030-60-90General
Side ratio1:1:√21:√3:2Varies
Legs equal?YesNoMaybe
Angles45°,45°,90°30°,60°,90°Varies
From square diagonal?YesNoNo

FAQ

What is the 45-45-90 triangle ratio?

leg : leg : hypotenuse = 1 : 1 : √2. If legs = 1, hypotenuse = √2 ≈ 1.414.

Why is it called isosceles right?

Isosceles = two equal sides (legs). Right = one 90° angle. The two equal angles must be 45° each.

How does the square diagonal relate?

A square's diagonal splits it into two 45-45-90 triangles. Diagonal = side × √2.

How do I find leg from hypotenuse?

leg = hypotenuse ÷ √2 = hypotenuse × √2/2. Or multiply by 0.7071 for a quick estimate.

What is the area formula?

Area = ½ × leg × leg = leg²/2. Half the area of a square with that side.

When to use in real life?

Construction (miter joints), design (45° rotations), navigation (equal N/E distances), engineering.

How to verify 45-45-90?

Check: 90° angle, two equal legs, hypotenuse = leg × √2.

Why is √2 irrational?

√2 cannot be written as a fraction. The Greeks proved this — a major discovery.

45-45-90 by the Numbers

1:1:√2
Side Ratio
45°
Base Angles
≈1.414
√2 Value
½ leg²
Area

Disclaimer: Results use standard floating-point precision. For critical construction or engineering, verify with professional tools.

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