Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the shape of the triangle of largest area that can be inscribed in a circle of radius \(r\). Hinti Let \(\alpha, \beta\), and \(\gamma\) be the central angles that subtend the three sides of the triangle. Show that the area of the triangle is \(\left.\frac{1}{2} r^{2} \mid \sin \alpha+\sin \beta-\sin (\alpha+\beta)\right]\). Maximize.

Step-by-Step Solution

Answer

The triangle of the largest area inscribed in the circle is equilateral.

Step 1: Understanding the Problem

First, identify the problem and necessary formula. We want to find the shape of a triangle with the largest area inscribed in a circle of radius \(r\). The area of the triangle inside the circle is given by: \(\frac{1}{2} r^{2} (\sin \alpha + \sin \beta - \sin (\alpha + \beta))\).

Step 2: Using Sine Rule

Use the sine rule for angles in a triangle inscribed in a circle. This rule relates the angles of the triangle to the radius of the circle. We have the identity: \(\alpha + \beta + \gamma = \pi\). Hence, \(\gamma = \pi - (\alpha + \beta)\).

Step 3: Simplify the Area Formula

Substitute \(\gamma = \pi - (\alpha + \beta)\) into the area formula. The area now is: \(A = \frac{1}{2}r^{2}(\sin \alpha + \sin \beta - (-\sin \gamma)) = \frac{1}{2}r^{2} (\sin \alpha + \sin \beta + \sin \gamma)\).

Step 4: Maximize Area Using Derivatives

To maximize this area, note that the function \(\sin \alpha + \sin \beta + \sin(\pi - (\alpha + \beta))\) reaches its maximum when all the angles are equal. Given that \(\alpha + \beta + \gamma = \pi\), this occurs when \(\alpha = \beta = \gamma = \frac{\pi}{3}\).

Step 5: Determine the Shape of the Triangle

Since \(\alpha = \beta = \gamma = \frac{\pi}{3}\), the triangle is equilateral. Therefore, the shape of the triangle of largest area that can be inscribed in the circle is an equilateral triangle.

Key Concepts

Triangle Area Maximization

Finding the shape of a triangle that maximizes its area when inscribed in a circle is an intriguing geometric problem. To tackle this, we use a formula that relates the area of the triangle to the central angles subtending its sides. This approach relies on the circle's radius, denoted by \( r \), and the sine of the angles. The formula expressing the area is:

• \( A = \frac{1}{2} r^{2} (\sin \alpha + \sin \beta + \sin \gamma) \)

Where \( \alpha, \beta, \) and \( \gamma \) are central angles corresponding to the triangle's sides.
This formula shows that maximizing the expression \( \sin \alpha + \sin \beta + \sin \gamma \) will, in turn, maximize the area of the triangle. An important mathematical step involves using derivatives to determine the maximum value of this expression.
Geometrically, this maximum is achieved when each of these central angles \( \alpha, \beta, \) and \( \gamma \) are equal, leading us to the surprising conclusion that the optimal triangle is equilateral. The intuition here is that symmetry balances the circle's division into equal segments, optimizing both the enclosed area and structural harmony.

Equilateral Triangle Properties

Equilateral triangles hold unique and fascinating properties that play a key role in solving problems like maximizing areas within circles. Each side of an equilateral triangle is the same length, and similarly, all internal angles are identical, measuring \(60^\circ\) or \(\frac{\pi}{3}\).
What makes an equilateral triangle special in the context of circle geometry is its symmetry. This symmetry implies several things:

• Equal central angles when inscribed in a circle.
• Optimized distribution of the angular gap among vertices, ensuring area maximization.
• Simplified calculations for analysts, thanks to uniformity across its geometric fields.

The harmonization within the equilateral triangle makes it an ideal candidate when tasks require consistent properties or evenly distributed forces, as in the case of the area maximization inside a circle. Whether studying integral calculus or physics, equilateral triangles often provide a clean, simple solution that leverages their inherent symmetry.

Central Angles in Circle Geometry

Understanding central angles is essential when solving geometric problems involving circles and inscribed figures. A central angle is an angle whose apex (vertex) is the center of the circle, and its arms extend to the circle's perimeter.
In our problem, central angles \( \alpha, \beta, \text{and} \gamma \) correspond to the sides of the triangle inscribed in the circle. By the nature of a circle's geometry:

• These angles sum to \( \pi \) (180 degrees) since they encompass half of the circle's available angular space.
• This constraint helps in redistributing circle segments uniformly, aiding in optimal structural configurations.
• When each central angle is equal, indicating \( \alpha = \beta = \gamma = \frac{\pi}{3} \), the triangle becomes equilateral.

The allocation and properties of central angles in a circle provide insight into achieving various geometric goals, be it constructing shapes with maximum area or studying circle-based discrete structures. By efficiently utilizing these angles, one can achieve harmonious and often unanticipated outcomes.