Problem 38 - Algebra and Trigonometry

Question

Find the area of the segment of a circle whose radius is 5 inches, formed by a central angle of \(40^{\circ}\).

Step-by-Step Solution

Answer

\[ \frac{25\pi}{9} - \frac{25}{2} \sin\left(\frac{2\pi}{9}\right) \text{ square inches} \]

Step 1: Convert Central Angle to Radians

To find the area of the segment, the central angle must be in radians. Convert the given angle from degrees to radians using the formula: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \] For a central angle of 40°, the conversion is: \[ 40^{\circ} \times \frac{\pi}{180} = \frac{40\pi}{180} = \frac{2\pi}{9} \text{ radians} \]

Step 2: Calculate Area of the Sector

The area of the sector is given by the formula: \[ A = \frac{1}{2}r^2\theta \] where \( r \) is the radius and \( \theta \) is the central angle in radians. Substituting \( r = 5 \) inches and \( \theta = \frac{2\pi}{9} \): \[ A = \frac{1}{2} \times 5^2 \times \frac{2\pi}{9} = \frac{1}{2} \times 25 \times \frac{2\pi}{9} = \frac{25\pi}{9} \text{ square inches} \]

Step 3: Calculate Area of the Triangle Formed

The triangle is formed by the two radii and the chord connecting the endpoints of the arc. The area of the triangle formed by the central angle \( \theta \) in a circle with radius \( r \) can be calculated as: \[ A_{\text{triangle}} = \frac{1}{2}r^2 \sin(\theta) \] Substitute \( r = 5 \) inches and \( \theta = \frac{2\pi}{9} \): \[ A_{\text{triangle}} = \frac{1}{2} \times 5^2 \times \sin\left(\frac{2\pi}{9}\right) = \frac{25}{2} \times \sin\left(\frac{2\pi}{9}\right) \]

Step 4: Final Calculation

Combine the areas calculated to find the area of the segment. The area of the segment is the area of the sector minus the area of the triangle: \[ A_{\text{segment}} = \frac{25\pi}{9} - \frac{25}{2} \sin\left(\frac{2\pi}{9}\right) \]

Key Concepts

Radian Conversion

Understanding how to convert degrees to radians is essential in many mathematical problems, especially when dealing with circles. The conversion is done using the formula:

• \(\text{radians} = \text{degrees} \times \frac{\pi}{180} \)

For example, to convert a central angle of 40° to radians:

• \(40^{\circ} \times \frac{\pi}{180} = \frac{2\pi}{9} \text{ radians} \)

This conversion is vital because many formulas for circle calculations use radians instead of degrees.

Circle Area Calculation

Calculating the area of a circle or a portion of it often involves knowing the radius and the central angle. The total area of a circle with radius \(r\) is given by:

• \(\text{Area} = \pi r^2\)

However, if we are dealing with a sector of the circle, we use the formula:

• \( A = \frac{1}{2}r^2\theta \)

Here, \( \theta \) is the central angle in radians. Using this, the area of a sector with a 40° angle and a radius of 5 inches becomes:

• \( A = \frac{1}{2} \times 5^2 \times \frac{2\pi}{9} = \frac{25\pi}{9} \text{ square inches} \)

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Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are crucial for solving problems involving triangles and circles. In our problem, the sine function helps determine the area of the triangle formed by the circle's radii and the chord. The area of such a triangle is calculated as:

• \( A_{\text{triangle}} = \frac{1}{2}r^2 \sin(\theta) \)

With \( r = 5 \) inches and \( \theta = \frac{2\pi}{9} \):

• \( A_{\text{triangle}} = \frac{25}{2} \times \sin\left(\frac{2\pi}{9}\right) \)

Trigonometric functions are necessary for various geometric and physical applications.

Geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and properties of figures. In this exercise, basic geometric principles help us understand the structure of the circle segment. A segment of a circle is a region bounded by a chord and an arc. Understanding the properties of radii, chords, and central angles is critical. These concepts include:

• A chord is a line segment whose endpoints lie on the circle.
• A central angle is an angle whose vertex is the center of the circle.

These basic principles lay the foundation for more advanced calculations, such as finding areas.

Circle Segment Area

The area of a circle segment is found by subtracting the area of the triangle formed by the radii and the chord from the area of the sector. To summarize, the steps are:

• Calculate the area of the sector \( \left( A = \frac{25\pi}{9} \right) \)
• Calculate the area of the triangle \( \left( A_{\text{triangle}} = \frac{25}{2} \sin\left( \frac{2\pi}{9} \right) \right) \)
• Subtract the triangle's area from the sector's area: \( A_{\text{segment}} = \frac{25\pi}{9} - \frac{25}{2} \sin\left( \frac{2\pi}{9} \right) \)

This gives the area of the segment, which encompasses all the key elements of trigonometry and geometry to find the solution.