Problem 252 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Compare the areas and perimeters of octagons which are respectively inscribed in and circumscribed to a given circle.

Step-by-Step Solution

Answer

The area and the perimeter of an octagon inscribed in a circle are always less than those of an octagon circumscribed to the same circle.

Step 1: Calculate the side length of the inscribed octagon

It is known that for an inscribed octagon, the side length \(s_i\) is given by \(s_i = 2r \sin(\pi/8)\) where r is the radius of the circumscribing circle.

Step 2: Calculate the side length of the circumscribed octagon

For a circumscribed octagon, the side length \(s_c\) is given by \(s_c = 2r \tan(\pi/8)\).

Step 3: Calculate the area and perimeter of the inscribed octagon

The area \(A_i\) and perimeter \(P_i\) of an inscribed octagon are given by \(A_i = 4r^2 \cos(\pi/8)\sin(\pi/8)\) and \(P_i = 8s_i\) respectively.

Step 4: Calculate the area and perimeter of the circumscribed octagon

The area \(A_c\) and perimeter \(P_c\) of a circumscribed octagon are given by \(A_c = 4r^2 \cot(\pi/8)\) and \(P_c = 8s_c\) respectively.

Step 5: Compare the areas and perimeters

Now compare \(A_i\) with \(A_c\) and \(P_i\) with \(P_c\) to note that the area and the perimeter of the octagon inscribed in a circle are always less than those of the octagon circumscribed to the same circle.

Key Concepts

Inscribed and circumscribed shapes

In geometry, inscribed and circumscribed shapes play a fascinating role in understanding the relationships and properties between different figures. An inscribed shape is one that fits snugly inside another shape, with all the vertices lying on the boundary of the larger shape. When we talk about an inscribed octagon, it means that the eight-sided polygon is drawn inside a circle with each vertex touching the circle.

On the other hand, a circumscribed shape surrounds another, with its sides tangent to the boundary of the enclosed shape. For a circumscribed octagon, the circle would be inside the octagon, tangential to each of the octagon's sides. This duality between inscribed and circumscribed shapes is crucial for comparing their areas and perimeters.

• Inscribed octagon: Every corner (vertex) touches the circle.
• Circumscribed octagon: The circle touches each side of the octagon once.

This concept helps in calculations in geometry as it determines the positioning and measurement strategies used for lengths and angles.

Trigonometry in geometry

Trigonometry is an essential tool in geometry, especially when dealing with polygons like octagons inscribed or circumscribed around circles. For inscribed shapes, trigonometry allows you to determine the side lengths based on the radius of the circle. For example, an octagon inscribed in a circle utilizes the sine function to find its side length: \(s_i = 2r \sin(\pi/8)\).

Similarly, trigonometry also helps in finding the side length of circumscribed shapes. For a circumscribed octagon, the tangent function is used: \(s_c = 2r \tan(\pi/8)\). Here, the relationships of angles within the circle and octagon guide us in computing the areas and perimeters of these figures.

• Inscribed octagon side length uses \(\sin\).
• Circumscribed octagon side length uses \(\tan\).

By understanding these trigonometric concepts, students can solve geometric problems that involve complex shapes by breaking them down into manageable parts.

Octagons

An octagon is a polygon with eight sides and eight angles. In geometry, octagons are interesting because they combine symmetry with complexity, making them ideal for mathematical exploration of areas and perimeters. When an octagon is inscribed or circumscribed around a circle, it forms a regular octagon, meaning all sides and angles are equal.

For an octagon inscribed in a circle, the formula for the area involves both cosine and sine functions: \(A_i = 4r^2 \cos(\pi/8)\sin(\pi/8)\). This is because each triangular section within the octagon takes advantage of those trigonometric ratios.
For a circumscribed octagon, the area is given by \(A_c = 4r^2 \cot(\pi/8)\), reflecting the different geometry of its position around the circle.

• Octagons inscribed in circles leverage\(\cos\) and \(\sin\).
• Circumscribed octagons use \(\cot\) to find the area.

With this engagement with octagons, students can see how different positioning relative to a circle changes the geometric properties and the calculations needed.