Calculating the Radius of a Square

You probably think of a radius as a property of a circle in two dimensions or of a three-dimensional sphere. However, mathematicians also use the term to refer to certain distances in regular polygons. In more casual use, radius of a square may also refer to the radius of a circle associated with the square in question.

Use of the Term Radius for Polygons

The radius of a regular polygon, such as a square, pentagon or octagon, is the distance from the center of the polygon to any of its vertices. Although this is proper usage of the word “radius,” it is rare to hear it used this way in practice. It is most often used for its more common meaning as the distance from the center of a circle to the circumference.

Calculating Radius of a Square

The distance from the center of a square to any one of its four corners can be calculated by taking half the length of one side of the square, squaring that value, doubling the result, then taking the square root of that number.

For example, for a 6-inch square (each side is 6 inches):

(text{Half of } 6 = frac{6}{2}= 3 3^2 = 3 × 3 = 9 \text{Doubling } 9 = 2 × 9 = 18 \sqrt{18} = 4.24)

The radius of a 6-inch square is 4.24 inches.

Pythagorean Theorem

The calculation for the radius of a square relies on the Pythagorean Theorem that describes the relationships of the sides of a right triangle:

(a^2 + b^2 = c^2)

The radius of the square is ​c​, the hypotenuse of a right triangle with sides, ​a​ and ​b​, that are half the length of the side of the square. The steps for calculating the radius derive directly from this formula.

Calculating Radius of an Inscribed Circle

For a circle in a square that just touches the edges of the square, the radius of the circle is one-half the length of the side of the square. For a 2-inch square, the radius of the circle is one inch.

Calculating Radius of a Circumscribed Circle

For a circle on the outside of the square that passes through all the vertices, known as a circumscribed circle, the radius of the circle is identical to the radius of the square. For a 2-inch square, the radius of the circle is 1.414 inches.

TL;DR (Too Long; Didn’t Read)

The term “radius,” while technically correct when applied to a square or another regular polygon, is rarely used except for circles.