What is the radius of a square?

The radius of a square is a fundamental geometric measurement defined as the distance from the square's central point to any of its four vertices (corners). It is precisely equivalent to one-half of the length of the square's diagonal.

Understanding the Concept of a Square's Radius

While the term "radius" is most commonly associated with circles, it can also be applied to regular polygons like a square. When we talk about the radius of a square, we are referring to a specific line segment within its structure.

Specifically, the radius of a square is the line segment that runs from the center of the square to any of its vertices. Imagine drawing a line from the very middle of the square straight out to one of its corners; that line represents the square's radius. This concept is closely related to the circumcircle of the square, which is the circle that passes through all four of its vertices, with its center coinciding with the square's center. The radius of this circumcircle is exactly the square's radius.

Formula for the Radius of a Square

To calculate the radius (r) of a square, you primarily need to know the length of its diagonal (d) or its side length (s).

Using the Diagonal (d):
The most direct way to find the radius is by taking half of the diagonal's length.
r = d / 2
Using the Side Length (s):
If you only know the side length of the square, you first need to calculate the diagonal. For a square, the diagonal (d) can be found using the Pythagorean theorem, where d = s√2.
Substituting this into the radius formula:
r = (s√2) / 2

Practical Examples and Calculations

Let's illustrate how to calculate the radius with a couple of examples.

Example 1: Given the Diagonal
- Problem: A square has a diagonal measuring 14 centimeters. What is its radius?
- Solution:
  - Using the formula r = d / 2
  - r = 14 cm / 2
  - r = 7 cm
  - The radius of the square is 7 centimeters.
Example 2: Given the Side Length
- Problem: A square has a side length of 8 inches. What is its radius?
- Solution:
  - First, find the diagonal: d = s√2 = 8√2 inches.
  - Next, find the radius: r = d / 2 = (8√2) / 2 = 4√2 inches.
  - Approximately, 4√2 ≈ 4 × 1.414 = 5.656 inches.
  - The radius of the square is approximately 5.66 inches.

Key Characteristics of a Square's Radius

Understanding these characteristics helps to solidify the concept:

It originates from the geometric center of the square.
It extends outwards to any of the square's four vertices.
Its length is precisely half the length of the square's diagonal.
It is also the radius of the unique circle that can be drawn around the square, touching all its corners.

Summary of Square Measurements

Here's a quick reference table summarizing the relationships between a square's side, diagonal, and radius:

Measurement	Definition	Formula (using side 's')	Formula (using diagonal 'd')
Side (s)	Length of any edge	`s`	`d / √2`
Diagonal (d)	Line connecting opposite vertices	`s√2`	`d`
Radius (r)	From center to any vertex	`(s√2) / 2`	`d / 2`