Problem 254 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If an equilateral triangle and a regular hexagon have the same perimeter, prove that their areas are as \(2: 3\).

Step-by-Step Solution

Answer

When an equilateral triangle and a regular hexagon have the same perimeter, their areas are indeed in the ratio of \(2:3\). This is because the area of each figure is proportionally related to the square of its side length (a), and their side lengths are in the ratio \(3:6\), or \(1:2\), given their perimeters are equal.

Step 1: Expression of the Perimeters

The problem statement provides that an equilateral triangle and a regular hexagon share the same perimeter. Let's say that the length of a side in both shapes is 'a'. As per the definitions, an equilateral triangle's perimeter is \(3a\) and a hexagon's perimeter is \(6a\). So we have \(3a = 6a\).

Step 2: Calculation of Areas

The next step is to calculate the area of the equilateral triangle and the regular hexagon. Area of an equilateral triangle = \(\sqrt{3}a^2/4\) and area of a regular hexagon = \((3\sqrt{3}a^2)/2\).

Step 3: Setting ratio of Areas

The last step is to find the ratio of the areas of the equilateral triangle and the hexagon. Simplifying \(\sqrt{3}a^2/4: (3\sqrt{3}a^2)/2\) we find the ratio to be \(2:3\). This shows that the area of the equilateral triangle is 2/3 the area of the regular hexagon.

Key Concepts

Understanding Geometry

Geometry, the branch of mathematics concerned with the properties and relationships of points, lines, shapes, and solids, is fundamental to understanding the world around us. It allows us to analyze and comprehend various forms and figures, providing a framework for problem-solving that applies to numerous fields including art, architecture, engineering, and science.

Among various geometric concepts, the relationship between the perimeter and area of shapes like equilateral triangles and regular hexagons is especially interesting, as it showcases how fixed parameters can yield different spatial properties. By exploring these relationships, learners can gain deeper insights into the principles that govern geometry.

Exploring the Equilateral Triangle

An equilateral triangle is one of the simplest and most symmetric forms in geometry. It is a polygon with three equal sides and three equal angles, each measuring 60 degrees. Due to its uniformity, an equilateral triangle exhibits interesting properties such as having concurrent angle bisectors, altitude lines, and medians.

Calculation of the Perimeter and Area

From a mathematical standpoint, the perimeter of an equilateral triangle is as straightforward as multiplying the length of a side, represented as 'a', by three. Its area, however, involves a more intricate formula, \( \sqrt{3}a^2 / 4 \), derived from the fact that each side can be considered as the base of a triangle comprising an altitude that bisects it.

Diving into the Regular Hexagon

On another note, a regular hexagon, also highly symmetrical, boasts six equal sides and angles. Geometry students learn that this six-sided polygon can be subdivided into equilateral triangles, which makes understanding its properties a little easier once they are familiar with the attributes of an equilateral triangle.

Understanding Perimeter and Area

The perimeter of a regular hexagon is six times the length of one side, and the area is represented by the formula \( (3\sqrt{3})a^2 / 2 \), demonstrating a clear connection to the equilateral triangles it encompasses.

The Intricacies of Area Calculation

Area calculation is an essential skill in geometry that involves determining the size of a two-dimensional surface enclosed within a set of lines. Each geometrical shape has its unique formula for calculating its area, which is a direct application of its defining properties.

For triangles and hexagons with equal perimeters, applying these formulas reveals a fixed ratio between their areas, regardless of the actual length of their sides. This demonstrates the power of mathematical relationships and underscores the importance of mastering area formulas to solve geometrical problems.

The Role of Mathematical Proof

Mathematical proof is a logical argument that confirms the truth of a geometrical statement. It is a step-by-step process that uses definitions, postulates, propositions, and previously proven theorems to arrive at a conclusion.

In our context, the proof involves showing that if an equilateral triangle and a regular hexagon have the same perimeter, the ratio of their areas is always 2:3. The proof uses the formulas for the area of each shape and establishes the ratio by simplifying the respective expressions. Such proofs are fundamental in geometry, as they provide certainty and solidify understanding of conceptual relationships.