Problem 1 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The angle of elevation of the top of a vertical pole when observed from each vertex of a regular hexagon is \(\frac{\pi}{3}\). If the area of the circle circumscribing the hexagon be \(A\) sq. metre, then the area of the hexagon is (A) \(\frac{3 \sqrt{3}}{2} A\) sq. metres (B) \(\frac{3 \sqrt{3}}{\pi} A\) sq. metres (C) \(\frac{3 \sqrt{3}}{2} \frac{A}{\pi}\) sq. metres (D) none of these

Step-by-Step Solution

Answer

Option C: \(\frac{3\sqrt{3}}{2} \frac{A}{\pi}\) sq. metres.

Step 1: Understanding the Problem

We know that the angle of elevation from any vertex of the hexagon to the top of the pole is \(\frac{\pi}{3}\). This implies that the altitude of the pole and the distance from any vertex to the base of the pole forms an equilateral triangle. We need to find the area of the regular hexagon when given the area of the circumscribed circle.

Step 2: Determine Key Relationships

The regular hexagon can be divided into 6 equilateral triangles. If \(s\) is the side length of each triangle, then the radius \(R\) of the circumscribed circle is equal to \(s\). The area \(A\) of the circle is \(A = \pi R^2\).

Step 3: Calculate Side Length of Hexagon

Since \(R = s\), and the radius equals the side length of the hexagon, we have \(R^2 = s^2\).

Step 4: Find Hexagon Area in terms of R

The area of the regular hexagon is given by \( \frac{3\sqrt{3}}{2}s^2 \). Replacing \(s^2\) with \(R^2\), the expression becomes \( \frac{3\sqrt{3}}{2}R^2 \). Since \(R^2 = \frac{A}{\pi}\), we have the hexagon's area as \( \frac{3\sqrt{3}}{2} \times \frac{A}{\pi} \).

Step 5: Select the Correct Option

The expression for the area of the hexagon is \( \frac{3\sqrt{3}}{2} \times \frac{A}{\pi} \), which corresponds to option (C).

Key Concepts

Regular Hexagon

A regular hexagon is a polygon with six equal sides and six equal angles. Each interior angle of a regular hexagon is 120 degrees, and each exterior angle is 60 degrees.
One of the unique properties of a regular hexagon is that it can be divided into six equilateral triangles.
This can be easily visualized if you draw straight lines connecting each vertex to the center of the hexagon.

• The side length of the hexagon is consistent throughout. This means each side measures the same distance.
• Each of the equilateral triangles formed internally will have three angles of 60 degrees.

These characteristics allow us to use the properties of equilateral triangles to solve for various measurements like the hexagon's area or side length, further emphasizing the interconnected nature of polygonal geometry.

Angle of Elevation

The concept of the angle of elevation is particularly useful in trigonometry and geometry when dealing with heights and distances. The angle of elevation from a point is the angle formed between the horizontal and the line of sight to a higher point.
In the exercise, each vertex of the hexagon at ground level forms this angle with the top of a vertical pole.

• The given \angle is \(\frac{\pi}{3}\), which corresponds to 60 degrees.
• This implies that the triangle formed between the pole, a point on the ground, and the base is an equilateral triangle (as \angle equals in all).

This angle helps calculate real-world distances and, in this case, assists in concluding that the radius to the top of the pole creates equilateral triangles. Thus, it directs us towards determining the relationships required to solve the problem.

Circumscribed Circle

In geometry, a circumscribed circle of a polygon, also known as a circumcircle, is a circle that passes through all the vertices of the polygon.
For regular polygons like a regular hexagon, the circumcircle is particularly symmetrical, and its center coincides with the center of the polygon.

• The radius of the circumcircle, often denoted as \(R\), is the distance from the circle's center to any of its vertices.
• In the case of a regular hexagon, this radius is equal to the side length of the hexagon itself.

Understanding the relationship between the circumcircle and the hexagon is crucial, as it allows calculations of the side length or the area of the hexagon when given the area of the circle. This connection simplifies solving complex geometric problems by creating a bridge between linear and circular measures.