This article popularly explains how to find the radius of a circle inscribed in a square. Theoretical material will help you understand all the nuances related to the topic. After reading this text, you can easily solve similar problems in the future.
Basic theory
Before proceeding directly to finding the radius of the circle inscribed in the square, it is worth familiarizing yourself with some fundamental concepts. They may seem too simple and obvious, but they are necessary for understanding the issue.
A square is a quadrangle, all sides of which are equal, and the degree measure of all angles is 90 degrees.
A circle is a two-dimensional closed curve located at a certain distance from a certain point. A segment, one end of which lies in the center of the circle, and the other on any surface, is called the radius.
We familiarized ourselves with the terms, only the main question remained. We need to find the radius of the circle inscribed in the square. But what does the last phrase mean? Nothing complicated here either. If all sides of a polygon touch a curved line, then it is considered inscribed in this polygon.
The radius of the inscribed circle
With theoretical material finished. Now you need to understand how to put it into practice. We use the drawing for this.
The radius is obviously perpendicular to AB. This means that at the same time it is parallel to AD and BC. Roughly speaking, you can "impose" it on the side of the square to further determine the length. As you can see, the segment BK will correspond to it.
One of its ends r lies in the center of the circle, which is the intersection point of the diagonals. The latter, according to one of their properties, divide each other in half. Using the Pythagorean theorem, we can prove that they also divide the side of the figure into two identical parts.
Accepting these arguments, we conclude:
r = 1/2 × a.