Problem 8 - Basic Engineering Mathematics

Question

Calculate the areas of the following sectors of circles: (a) having radius \(6 \mathrm{~cm}\) with angle subtended at centre \(50^{\circ}\) (b) having diameter \(80 \mathrm{~mm}\) with angle subtended at centre \(107^{\circ} 42^{\prime}\) (c) having radius \(8 \mathrm{~cm}\) with angle subtended at centre \(1.15\) radians.

Step-by-Step Solution

Answer

Areas: (a) ≈ 15.71 cm², (b) ≈ 15.076 cm², (c) = 36.8 cm².

Step 1: Understand the Formula for the Area of a Sector

The area of a sector of a circle can be calculated using the formula: \( A = \frac{\theta}{360^{\circ}} \times \pi r^2 \) when the angle \( \theta \) is in degrees, or \( A = \frac{1}{2} r^2 \theta \) when \( \theta \) is in radians. Here, \( r \) is the radius of the circle and \( \pi \) is approximately 3.14159.

Step 2: Solve for part (a)

For (a), the radius \( r = 6 \text{ cm} \) and \( \theta = 50^{\circ} \). Substituting these values into the formula for the area of a sector in degrees, we get:\[A = \frac{50}{360} \times \pi \times 6^2\]Simplifying:\[A = \frac{50}{360} \times \pi \times 36 = \frac{5}{36} \times 36 \pi = 5 \pi \approx 15.71 \text{ cm}^2\]

Step 3: Solve for part (b)

For (b), first convert the given angle into degrees. The angle is \(107^{\circ} 42^{\prime}\), which can be calculated as \(107 + \frac{42}{60} = 107.7^{\circ}\). The diameter is 80 mm, thus the radius \( r = 40 \text{ mm} \) or \( 4 \text{ cm} \).Substitute these into the sector area formula:\[A = \frac{107.7}{360} \times \pi \times 4^2\]Simplifying:\[A = \frac{107.7}{360} \times \pi \times 16 \approx 15.076 \text{ cm}^2\]

Step 4: Solve for part (c)

Here the radius \( r = 8 \text{ cm} \) and \( \theta = 1.15 \text{ radians}\). Using the formula for the area with \( \theta \) in radians:\[A = \frac{1}{2} \times 8^2 \times 1.15\]Simplify:\[A = \frac{1}{2} \times 64 \times 1.15 = 36.8 \text{ cm}^2\]

Step 5: Final Step: Conclude the Areas of the Sectors

Based on the calculations:- For part (a), the area is approximately \( 15.71 \text{ cm}^2 \).- For part (b), the area is approximately \( 15.076 \text{ cm}^2 \).- For part (c), the area is \( 36.8 \text{ cm}^2 \).

Key Concepts

Area of Sectors

The area of a sector in a circle can be likened to a slice of a pizza. Imagine how a slice might look differently if the pizza is larger, or if the slice is wider. That's fundamentally what influences the area of sectors — the size of the circle and the size of the angle at the circle's center.

• To calculate this area, you use the radius of the circle and the angle of the slice, known as the central angle.


• If the angle is in degrees, the formula used is: \( A = \frac{\theta}{360^{\circ}} \times \pi r^2 \).


• If the angle is in radians, the formula changes slightly: \( A = \frac{1}{2} r^2 \theta \).


• These formulas essentially measure how big a portion of the whole circle is being discussed.

The calculations help you determine how large or small these "slices" are when compared to the whole circle. This idea may also be applied to problems involving circular objects, such as pizzas or pie charts in statistics.

Circle Geometry

Circle geometry deals with the properties and applications of circles. In this context, we focus on sectors—parts of a circle's area that are enclosed by two radii and an arc. Key elements in circle geometry include:

• Radius: The distance from the circle's center to any point on its boundary. It's a key measurement in the area of a sector.


• Diameter: Double the radius, it extends across the circle's widest point and is sometimes given in problems to find the radius (as seen in problem part b).


• Central Angle: This is the angle formed at the circle's center by the two radii that form a sector; it's crucial for determining the "size of the slice."

Understanding circle geometry is vital for solving problems related to any circular shapes and involves some visualizing and measuring, making it a practical, tangible field of mathematics.

Radians and Degrees

Radians and degrees are units for measuring angles, and they are vital in working with circle geometry. Knowing how to convert between these two is essential because formulas may require the angle to be in a specific unit.

• Degrees: Most familiar, split the angle into 360 parts, like slices of a very big pie. For instance, a complete circle goes around itself 360 degrees.


• Radians: These measure angles based on the radius of the circle. A full circle is \(2\pi\) radians because the circumference of a circle is \(2\pi\) times its radius.
• Conversion: To switch between them, use \(180^{\circ} = \pi\) radians, such as to convert degrees to radians multiply degrees by \(\frac{\pi}{180}\).

Understanding these units and the ability to convert between them are critical for mathematical competence and solving problems accurately.

Mathematical Formulas

Mathematical formulas are the backbone of calculations, providing a structured way to solve problems and find answers accurately. In the context of the exercise, specific formulas used play a key role.

• Sector Area Formula: There are distinct formulas depending on whether the angle is measured in degrees or radians. Degrees use \( A = \frac{\theta}{360^{\circ}} \times \pi r^2 \) and radians use \( A = \frac{1}{2} r^2 \theta \).


• Conversion Formulas: Techniques to convert angles, or measure other circle properties, are essential and rely heavily on formulas. These ensure you use consistent units across all calculations.


• Application: Using formulas correctly means understanding what each symbol stands for and ensuring all measures are consistent (like keeping units the same between radius and angle).

In mathematics, especially in problem-solving scenarios, mastering formulas is like having a set of reliable tools for building your answer accurately.