Diagonal of a Square Calculator
Calculate square diagonal length using side, area, or perimeter with step-by-step solutions
Calculate Square Diagonal
Length of one side of the square
Square Calculation Results
Example Calculation
Square Tile Example
Problem: Square ceramic tile
Side Length: 30 cm
Find: Diagonal length for cutting purposes
Application: Construction and tiling
Calculation Steps
Given: Side = 30 cm
Formula: d = a × √2
d = 30 × √2
d = 30 × 1.4142
d = 42.426 cm
Square Properties
Equal Sides
All four sides are equal in length
Right Angles
All four corners are 90°
Equal Diagonals
Both diagonals are equal and perpendicular
Symmetry
4 lines of symmetry, rotational symmetry
Formula Reference
Diagonal from Side
d = a × √2
Side from Diagonal
a = d / √2
From Area
d = √(2A)
From Perimeter
d = P√2/4
Key Constant
√2 ≈ 1.4142
Understanding Square Diagonals
What is a Square Diagonal?
A square diagonal is a line segment that connects two opposite corners (vertices) of a square. Every square has two diagonals that are equal in length, perpendicular to each other, and bisect each other at the center of the square.
Why Calculate the Diagonal?
- •Cutting materials diagonally
- •Finding distances across square areas
- •Designing square patterns and layouts
- •Solving geometry and trigonometry problems
Mathematical Foundation
d = a × √2
(Special case of Pythagorean Theorem)
The diagonal formula comes from the Pythagorean theorem. A square diagonal creates two 45-45-90 right triangles, where the diagonal is the hypotenuse and the sides are the legs.
Key Relationships
Diagonal to side ratio: √2 : 1 ≈ 1.414 : 1
Area from diagonal: A = d²/2
Circumradius: R = d/2
Inradius: r = a/2 = d/(2√2)
Real-World Applications
- • Square tile cutting and installation
- • Picture frame diagonal measurements
- • Construction and carpentry
- • Fabric cutting for quilts
- • Garden and landscape design
- • Engineering and technical drawings
Special Properties
- • Diagonals bisect each other at 90°
- • Each diagonal divides square into 2 triangles
- • Diagonal creates 45° angles with sides
- • Square inscribed in circle: d = diameter
- • Square circumscribed around circle: a = diameter
Related Concepts
- • 45-45-90 right triangles
- • Pythagorean theorem
- • Regular polygon properties
- • Circle inscribed/circumscribed
- • Coordinate geometry
- • Geometric transformations