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45-45-90 Triangle Calculator

45-45-90 Triangle Calculator

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

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45-45-90 Triangle Calculator
Enter one side of a 45-45-90 isosceles right triangle to find all other measurements using the ratio a : a : a√2
a:a:a2=1:1:2a : a : a\sqrt{2} = 1 : 1 : \sqrt{2}

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

First leg of isosceles right triangle

Second leg = a (equal legs)

Length = a√2

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
a = b = 1
c = ?

Unit 45-45-90 triangle

Example
a = b = 5
c = ?

Simple leg example

Example
a = b = ?
c = 10

Given hypotenuse

Example
a = b = 3
c = ?

Small triangle example

What Is a 45-45-90 Triangle?

A 45-45-90 triangle is an isosceles right triangle with angles of 45°, 45°, and 90°. It has two equal legs and a fixed side ratio that makes calculations straightforward.

Key Properties:

  • Angles: 45°, 45°, 90° (two equal acute angles)
  • Side Ratio: 1 : 1 : √2 (leg : leg : hypotenuse)
  • Equal Legs: Both legs (a and b) are equal in length
  • Hypotenuse: Always equals leg × √2
  • Isosceles: Two equal sides and two equal angles

Origin: This triangle appears when you draw a diagonal across a square, dividing it into two congruent 45-45-90 triangles.

How to Use the 45-45-90 Triangle Ratio

Calculation Method:

Given leg (a or b):
a=b=given lega = b = \text{given leg}
c=a2c = a\sqrt{2}
Given hypotenuse (c):
a=b=c2=c22a = b = \frac{c}{\sqrt{2}} = \frac{c\sqrt{2}}{2}

Area Formula:

Area=12×a×b=12×a2\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 √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
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):

a2+b2=c2a^2 + b^2 = c^2
Since a = b: a2+a2=c2a^2 + a^2 = c^2
2a2=c22a^2 = c^2
c=a2c = a\sqrt{2}
Therefore: a : a : a√2 = 1 : 1 : √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 = √(1² + 1²) = √2
• Triangle ratio: 1 : 1 : √2 ✓
• Each angle: 45°, 45°, 90° ✓

Explore more special triangles

30-60-90 TrianglePythagorean Theorem