Problem 248 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the length of the side of a regular polygon of 12 sides which is circumscribed to a circle of unit radius.

Step-by-Step Solution

Answer

The length of the side of the regular polygon circumscribed to a circle of unit radius is \(\sqrt{3}/2\) units.

Step 1: Understanding The Problem

A circle of radius 1 is given. A regular polygon of 12 sides is inscribed in the circle. All angles at the center for all those shapes are equal and sum up to \(360^\circ\). Therefore, each angle measures \(360^\circ / 12 = 30^\circ\). If you draw two radii which define one side of the polygon, a triangle is formed. It's an isosceles triangle because two sides are radii of the circle. Notice that \(30^\circ\) is half the angle at the center of the regular pentagon. Therefore, our triangle is a 30-60-90 triangle.

Step 2: Use the Properties of a 30-60-90 Triangle

In a 30-60-90 right triangle the sides are in the ratio 1: \(\sqrt{3}\): 2. Here the side opposite 30º is half the Hypotenuse (radius = 1) and other side which is also a side of a regular polygon of 12 sides is the side opposite 60º which equals Hypotenuse times cos 30º.

Step 3: Calculation

So, the required side length of the polygon will be 1 * cos(30º) = \(\sqrt{3}/2\) units.

Key Concepts

Geometry of Regular Polygons

A regular polygon is a geometric figure with all sides of equal length and all interior angles of equal measure. In geometry, understanding regular polygons is crucial as they provide the foundation for various geometric concepts and problem-solving techniques.

For instance, a 12-sided polygon, known as a dodecagon, can be inscribed in a circle which means all its vertices lie on the circle's circumference. When a regular polygon is inscribed in a circle, this circle is called the circumcircle. In the given exercise, the radius of the circumcircle is specified as 1.Due to the equality of the sides and angles of a regular polygon, we can divide it into congruent isosceles triangles by drawing lines (radii) from the center to each vertex. The central angle formed at the vertex where two radii meet is easily calculated by dividing 360 degrees over the number of sides, providing a methodical approach to solving related geometric problems.

Properties of 30-60-90 Triangles

The 30-60-90 triangle is a special type of right-angled triangle where the angles are in a fixed ratio of 1:2:3, meaning the angles are always 30 degrees, 60 degrees, and 90 degrees respectively. The sides of this triangle are also in a consistent ratio: the length of the side opposite the 30-degree angle is usually denoted as x, the length of the side opposite the 60-degree angle is x\(\times\sqrt{3}\), and the hypotenuse, opposite of the 90-degree angle, is 2x.

This knowledge is extremely useful when dealing with regular polygons, as each segment drawn from the centre to a vertex, along with a side of the polygon, forms a 30-60-90 triangle. With this, using simple trigonometric ratios, one can quickly determine the lengths of the sides involved. Furthermore, remembering these properties allows one to solve a variety of geometry problems with greater ease and accuracy.

Trigonometry in Polygons

Trigonometry is an essential tool when analyzing polygons, especially in the context of finding side lengths, angle measures, and areas. When a regular polygon is circumscribed about a circle, trigonometric functions such as sine, cosine, and tangent can be used in conjunction with the central angles to find the length of the sides.

In particular, cosine is often used to relate the radius of the circumcircle (often the hypotenuse in our 30-60-90 triangles) to the length of the side of the polygon. For a 12-sided polygon like a dodecagon, each central angle is 30 degrees. By applying the cosine function to this angle, we determine the length of the adjacent side (a side of the dodecagon), which is the required length in our exercise. This application of trigonometry simplifies the process and provides a clear path to the solution.