Problem 91 - Precalculus

Question

Find the area of the circular sector given the indicated radius and central angle. Round your answers to three significant digits. $$\theta=56^{\circ}, r=4.2 \mathrm{cm}$$

Step-by-Step Solution

Answer

The area of the circular sector is approximately 8.62 cm².

Step 1: Convert the Angle to Radians

To find the area of a circular sector, we first need to ensure the angle is in radians. The conversion formula from degrees to radians is: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). So, convert \(56^{\circ}\) as follows: \( \theta = 56 \times \frac{\pi}{180} = \frac{56\pi}{180} \approx 0.977\) radians.

Step 2: Use the Sector Area Formula

The formula for the area of a sector is \( A = \frac{1}{2} r^2 \theta \), where \( r \) is the radius and \( \theta \) is the angle in radians. Substitute \( r = 4.2 \) cm and \( \theta = 0.977 \) radians into the formula: \( A = \frac{1}{2} \times (4.2)^2 \times 0.977 \).

Step 3: Calculate the Area of the Sector

Calculate the expression from Step 2: \( A = \frac{1}{2} \times 4.2^2 \times 0.977 = \frac{1}{2} \times 17.64 \times 0.977 \). First, compute \( 4.2^2 = 17.64 \), then \( 17.64 \times 0.977 \approx 17.23968 \), and finally, \( \frac{17.23968}{2} \approx 8.61984 \).

Step 4: Round the Result

Round the area to three significant digits. The rounded area is approximately \( 8.62 \) square centimeters.

Key Concepts

Angle Conversion

Angle conversion is crucial when dealing with circular sectors. Usually, angles are provided in degrees, but many mathematical formulas, including the sector area formula, require angles in radians. To convert an angle from degrees to radians, you can use the conversion formula:

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

For example, in the original problem, the angle given is \( 56^{\circ} \). Using the formula, we obtain \( 56 \times \frac{\pi}{180} \approx 0.977 \) radians. This conversion helps in conveniently using the angle in the sector area formula.

Radian Measure

Radian measure is a way of expressing angles based on the radius of a circle. A radian is defined as the angle created when the arc length is equal to the radius of the circle. This is different from degrees, where a full circle is divided into 360 equal parts. Instead, in radians, a full circle equals \( 2\pi \) radians.

• For context, \( \pi \) radians is equivalent to \( 180^{\circ} \).
• So, \( \frac{\pi}{2} \) radians represents a right angle or \( 90^{\circ} \).

By understanding radians, you can easily interchange angles in terms of radians or degrees, which is valuable for various mathematical applications, including finding the area of a sector.

Radius

The radius is the straight line distance from the center of a circle to its edge. It is an essential metric when calculating areas or other properties of circles and sectors. In any circle, the radius is consistent, meaning, if you know the radius, you can determine other features of the circle such as its diameter, area, and more.

• The diameter is twice the radius, \( d = 2r \).
• The circle’s area is found using \( \pi r^2 \).

For sectors, the radius is used in the formula to calculate the sector area. In the given exercise, the radius \( r \) is \( 4.2 \) cm.

Sector Area Formula

The sector area formula is used to find the area of a section of a circle, akin to a 'slice' or 'wedge'. The formula is derived based on the fraction of the circle represented by the central angle and the radius:

• The sector area \( A \) is given by the formula \( A = \frac{1}{2} r^2 \theta \).
• Here, \( r \) is the radius and \( \theta \) is the angle in radians.

For instance, using \( r = 4.2 \, \text{cm} \) and \( \theta \approx 0.977 \, \text{radians} \), we plug these into the formula, calculating \( A = \frac{1}{2} \times 4.2^2 \times 0.977 \). This simplifies to about \( 8.62 \) square cm when rounded to three significant figures.