What is the ratio of sides in a 45-45-90 triangle?

The side ratio is 1:1:√2 or equivalently a:a:a√2. If the legs are length 'a', the hypotenuse is a√2 ≈ 1.414a. Both legs are always equal because the two base angles are both 45°.

How do I find the hypotenuse of a 45-45-90 triangle?

Multiply the leg length by √2 (approximately 1.414). For example, if a leg is 5 units, the hypotenuse is 5√2 ≈ 7.07 units. This works because in a 45-45-90 triangle, hypotenuse = leg × √2.

How do I find the legs from the hypotenuse?

Divide the hypotenuse by √2 (or multiply by √2/2). For example, if the hypotenuse is 10 units, each leg is 10/√2 = 10√2/2 = 5√2 ≈ 7.07 units.

Where does the 45-45-90 triangle appear?

45-45-90 triangles appear when you cut a square diagonally in half. They're used in architecture, engineering, and design. The diagonal of any square creates two 45-45-90 triangles.

What is the area formula for a 45-45-90 triangle?

Area = (1/2) × leg × leg = leg²/2. Since both legs are equal, if leg = a, then Area = a²/2. If you know the hypotenuse c, Area = c²/4.

Geometry

45-45-90 Triangle Calculator

Solve 45-45-90 isosceles right triangles using the ratio $1 : 1 : \sqrt{2}$ . Find missing sides, area, and perimeter with step-by-step solutions and visual triangle diagrams.

100% FreeInstant ResultsVisual Diagram

Enter one side of a 45-45-90 isosceles right triangle to find all other measurements using the ratio

a : a : a\sqrt{2}

Enter to calculate, Esc to clear

a : a : a\sqrt{2} = 1 : 1 : \sqrt{2}

45-45-90 triangle side ratio (isosceles right triangle)

Leg (a) - opposite 45°

First leg of isosceles right triangle

Leg (b) - opposite 45°

Second leg = $a$ (equal legs)

Hypotenuse (c) - opposite 90°

Length = $a\sqrt{2}$

Units

Display Mode

Instructions: Enter any one side length. In a 45-45-90 triangle, both legs are equal. The legs will auto-sync when you enter one.

Try These Examples

Click on any example to automatically fill the calculator

Example

Unit 45-45-90 triangle

a = b: 1

c: ?

Example

Simple leg example

a = b: 5

c: ?

Example

Given hypotenuse

a = b: ?

c: 10

Example

Small triangle example

a = b: 3

c: ?

What Is a 45-45-90 Triangle?

A 45-45-90 triangle is the simplest special right triangle you'll run into. It's what you get when you slice a square diagonally — two equal legs and a hypotenuse that's always $\sqrt{2}$ times longer. Because both acute angles are 45°, the two legs are always the same length.

Key Properties:

Angles: 45°, 45°, 90° (two equal acute angles)
Side Ratio: $1 : 1 : \sqrt{2}$ (leg : leg : hypotenuse)
Equal Legs: Both legs (a and b) are equal in length
Hypotenuse: Always equals leg $\times \sqrt{2}$
Isosceles: Two equal sides and two equal angles

Origin: Draw a diagonal across any square and you've got two of these triangles. That's why they show up constantly in architecture and design.

How to Use the 45-45-90 Triangle Ratio

Calculation Method:

Given leg (a or b):

a = b = \text{given leg}

c = a\sqrt{2}

Given hypotenuse (c):

a = b = \frac{c}{\sqrt{2}} = \frac{c\sqrt{2}}{2}

Area Formula:

\text{Area} = \frac{1}{2} \times a \times b = \frac{1}{2} \times a^2

(Since both legs are equal: a = b)

Memory Tip: The hypotenuse of a 45-45-90 triangle is always the leg length multiplied by $\sqrt{2}$ (approximately 1.414).

Real-World Applications

Construction & Architecture

Square diagonal calculations
45° roof slopes and ramps
Corner bracing in construction
Tile cutting for diagonal patterns
Stair design with 45° angles

Engineering & Design

Mechanical joint design
CAD drawing and blueprints
Structural diagonal supports
Electronic circuit layouts
Optical component positioning

Mathematics & Education

Coordinate geometry problems
Distance calculations
Trigonometry foundations
Vector component calculations
Physics force decomposition

Understanding the 45-45-90 Triangle: Theory and Practice

Why the 1:1:√2 Ratio Exists

The 45-45-90 triangle emerges naturally when you cut a square diagonally in half. Because a square has four equal sides and four 90° angles, its diagonal creates two congruent isosceles right triangles. Each triangle inherits two sides from the square (the legs) and gets the diagonal as its hypotenuse.

Using the Pythagorean theorem with equal legs ( $a = b$ ), we derive: $a^2 + a^2 = c^2$ → $2a^2 = c^2$ → $c = a\sqrt{2}$ . This proves the ratio is always $1:1:\sqrt{2}$ , approximately 1:1:1.414.

Real-World Example 1: Square Room Diagonal Bracing

Scenario: You're installing diagonal wood bracing across a square room measuring 10 feet × 10 feet to reinforce the structure during renovation.

• Room dimensions: 10 ft × 10 ft (square)

• Each wall forms a leg:

a = b = 10

• Diagonal bracing length:

c = 10\sqrt{2} \approx 14.14

• Material needed: At least 15 ft of lumber (accounting for cuts)

Without the 1:1:√2 ratio, you'd need to measure the diagonal physically or use complex calculations. The formula saves time: just multiply the wall length by 1.414. This principle applies to any square structure—tile layouts, window frames, or foundation corners.

Real-World Example 2: 45° Roof Pitch Calculation

Scenario: A shed design calls for a 45° roof pitch with an 8-foot horizontal span (run).

• Horizontal run:

a = 8

• Vertical rise (45° means equal):

b = 8

• Rafter length:

c = 8\sqrt{2} \approx 11.31

• Per rafter material: 12 ft board (11.31 ft + overhang)

A 45° pitch creates a steep, symmetrical roof perfect for shedding snow. Carpenters rely on this ratio when cutting rafters—knowing that a 45° angle means rise equals run simplifies layout. For our 8 ft span, each rafter must be $8\sqrt{2}$ ft, or about 11 feet 4 inches.

Connection to the Pythagorean Theorem

Every 45-45-90 triangle is a special case of the Pythagorean theorem ( $a^2 + b^2 = c^2$ ). The constraint that $a = b$ transforms the general formula into $2a^2 = c^2$ . This relationship explains why the hypotenuse is always $\sqrt{2}$ times longer than each leg—it's the square root of the sum of two equal squared sides.

Verification Example: If legs are 3 cm each: $3^2 + 3^2 = 9 + 9 = 18$ . The hypotenuse squared is 18, so $c = \sqrt{18} = 3\sqrt{2} \approx 4.24$ cm. The ratio $3:3:4.24$ reduces to $1:1:1.414$ ✓

Learn More from Authoritative Sources

OpenStax Geometry: Special Right Triangles

Free textbook chapter covering 45-45-90 and 30-60-90 triangles with proofs and practice problems.

Khan Academy: 45-45-90 Triangles

Video tutorial explaining the ratio derivation with step-by-step examples.

LibreTexts Mathematics: Triangle Properties

Comprehensive mathematics library covering triangle geometry and trigonometry fundamentals.

Proofs of 45-45-90 Triangle Ratios

Pythagorean Theorem Proof

Starting with the Pythagorean theorem and the fact that both legs are equal (a = b):

a^2 + b^2 = c^2

Since a = b:

a^2 + a^2 = c^2

2a^2 = c^2

c = a\sqrt{2}

Therefore:

a : a : a\sqrt{2} = 1 : 1 : \sqrt{2}

✓

Square Diagonal Method

Consider a unit square (side length = $1$ ). Drawing the diagonal creates two 45-45-90 triangles:

• Square side length = $1$
• Each triangle leg = $1$
• Diagonal = $\sqrt{1^2 + 1^2} = \sqrt{2}$
• Triangle ratio: $1 : 1 : \sqrt{2}$ ✓
• Each angle: $45°$ , $45°$ , $90°$ ✓

Frequently Asked Questions

A 45-45-90 triangle is a special right triangle with two 45° angles and one 90° angle. It's also called an isosceles right triangle because both legs are equal in length.

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