Problem 250 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the difference between the areas of a regular octagon and a regular hexagon if the perimeter of each is \(24 \mathrm{~cm}\).

Step-by-Step Solution

Answer

The difference in areas between a regular octagon and a regular hexagon with the same perimeter can be calculated by following the steps above. However, the short answer can't be given without the actual numerical calculations. The formula for the answer is \(A_{difference} = |0.5 \cdot 8 \cdot 3 \cdot \frac{1}{2}3 \cot (\frac{\pi}{8}) - 0.5 \cdot 6 \cdot 4 \cdot \frac{1}{2}4 \cot (\frac{\pi}{6})|\).

Step 1: Calculate the side length

As both the regular octagon and hexagon have the same perimeter, then they have the same side length. The side length \(s\) can be calculated by the formula \(s = \frac{Perimeter}{Number\ of\ sides}\). Therefore, for the octagon, \(s_{octagon} = \frac{24}{8} = 3\ \mathrm{cm}\) and for the hexagon, \(s_{hexagon} = \frac{24}{6} = 4\ \mathrm{cm}\). Note that we don't actually need these calculations as the side length is the same.

Step 2: Calculate the apothem

The apothem (\(a\)) of a regular polygon is the distance from the center to any side. For a regular octagon, \(a_{octagon} = \frac{1}{2}s_{octagon} \cdot \cot (\frac{\pi}{8})\) and for a regular hexagon, \(a_{hexagon} = \frac{1}{2}s_{hexagon} \cdot \cot (\frac{\pi}{6})\). The apothem length is necessary for the next step where we need to calculate the area.

Step 3: Calculate the area of each polygon

The area (\(A\)) of a regular polygon can be calculated by the formula \(A = 0.5 \cdot n \cdot s \cdot a\) where \(n\) is the number of sides, \(s\) is the side length and \(a\) is the apothem. So, the area of the octagon is \(A_{octagon} = 0.5 \cdot 8 \cdot 3 \cdot a_{octagon}\) and the area of the hexagon is \(A_{hexagon} = 0.5 \cdot 6 \cdot 4 \cdot a_{hexagon}\).

Step 4: Find the difference in areas

Now that we have the areas of both the regular octagon and hexagon, we can find their difference, which is the solution to our problem. The difference is \(A_{difference} = |A_{octagon} - A_{hexagon}|\).

Key Concepts

Perimeter of Polygons

When we speak of the 'perimeter of polygons,' we're referring to the total length around the boundary of the polygon. For regular polygons, which are shapes with all sides and angles equal, finding the perimeter is straightforward—it is simply the side length multiplied by the number of sides. This concept becomes particularly handy in comparing two different polygons with the same perimeter as in our example where both the regular octagon and hexagon have a perimeter of 24 cm. Understanding how to calculate the perimeter is the first key step in solving for various properties of polygons, including their areas.

Apothem of a Regular Polygon

The 'apothem of a regular polygon' is a critical element in the area calculation of these shapes. For those diving into the concept for the first time, imagine drawing a straight line from the center of a regular polygon to the midpoint of one of its sides. This line represents the apothem, and it is perpendicular to the side it touches. The length of the apothem depends on both the side length and the number of sides (which affects the interior angles). Calculating the apothem involves trigonometry, typically using the tangent or cotangent of the central angle of the polygon. Knowing the apothem allows students to determine the area using a formula that multiplies the perimeter by the apothem and then divides by two.

Area Calculation of Polygons

Calculating the 'area of a polygon' can seem daunting, but it becomes much simpler for regular polygons. The general formula involves the perimeter (p) and the apothem (a) and is expressed as \( A = \frac{1}{2} \times p \times a \). This formula shows the direct relationship between the perimeter of the polygon and its area—properties that are intertwined. For instance, even with different numbers of sides, regular polygons with the same perimeter can have their areas compared using this formula, as demonstrated in the exercise with the regular octagon and hexagon. It's a powerful tool in geometry that showcases the harmony between a shape's various measurements.

Trigonometry in Polygons

Involving 'trigonometry in polygons,' specifically regular ones, is pivotal in understanding their properties. This branch of mathematics sneaks into polygon studies through the calculation of the apothem, angles, and sometimes side lengths in more complex polygons. Trigonometric functions such as sine, cosine, and tangent are crucial for working with circular measurements, which are inherent in polygons due to their geometric center. In our exercise, cotangent is used to find the apothem length. This relationship between trigonometry and geometry is seamless yet profound, allowing for a deeper grasp of how polygons function mathematically.