Problem 93 - Precalculus

Question

Find the area of the circular sector given the indicated radius and central angle. Round your answers to three significant digits. $$\theta=1.2^{\circ}, r=1.5 \mathrm{ft}$$

Step-by-Step Solution

Answer

The area of the sector is approximately 0.0236 square feet.

Step 1: Convert Angle to Radians

The central angle is given in degrees. Convert it to radians using the formula \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \).Substituting the given angle:\( 1.2^{\circ} \times \frac{\pi}{180} \approx 0.02094395 \text{ radians} \).

Step 2: Recall the Formula for Area of a Sector

The area \( A \) of a sector with radius \( r \) and central angle \( \theta \) in radians is given by: \( A = \frac{1}{2} r^2 \theta \).

Step 3: Substitute the Values into the Formula

Now, we substitute \( r = 1.5 \text{ ft} \) and \( \theta = 0.02094395 \text{ radians} \) into the formula:\( A = \frac{1}{2} \times (1.5)^2 \times 0.02094395 \approx 0.0236 \text{ square feet} \).

Step 4: Round the Result to Three Significant Digits

The area calculated is approximately \( 0.0236 \text{ square feet} \), rounded to three significant digits: \( 0.0236 \).

Key Concepts

Radians

When dealing with angles, the unit of radians is incredibly important for measuring how much of a circle an angle covers. Unlike degrees, which divide a circle into 360 parts, radians use the radius of the circle as the basis for division.
One full circle is equivalent to \(2\pi\) radians, which means that one radian is the angle formed when the arc length is equal to the radius.
This unit is especially useful in mathematics because it simplifies many formulas involving angles, such as those used in trigonometry and calculus.To convert an angle from degrees to radians, use the formula:

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

This formula arises because \(180^{\circ}\) is equal to \(\pi\) radians.
In the original problem, the central angle of \(1.2^{\circ}\) is converted to radians as follows:

• \(1.2^{\circ} \times \frac{\pi}{180} \approx 0.02094395\) radians

Central Angle

A central angle is an angle whose vertex is located at the center of a circle, extending outwards to the circumference. This kind of angle is fundamental when working with sectors of a circle, an important geometric shape.
It essentially determines the size of the sector and relates directly to how much of the circle the sector will "slice."
The measurement of a central angle can be expressed in degrees or radians. However, for mathematical calculations involving circle sectors, radians are often more precise and simplify calculations.When working with formulas, always ensure the angle is in radians.
This is critical because the formula for the area of a sector is dependent on this measure:

• \( A = \frac{1}{2} r^2 \theta \)

Converting a central angle to radians step is crucial, as was shown in the exercise where \(1.2^{\circ}\) was converted into \(0.02094395\) radians.

Circular Sector

A circular sector is like a "pizza slice" of a circle. It consists of a region bounded by two radii and the arc between them.
This shape is a significant element in geometry due to its application in various real-world and theoretical problems.The area of a circular sector can be determined using the formula:

• \( A = \frac{1}{2} r^2 \theta \)

Here:

• \(r\) represents the radius of the circle.
• \(\theta\) is the central angle in radians.

This formula smartly uses the relationship between the angle and the radius to ascertain how much space the sector takes up.In the given exercise, with a radius of \(1.5\) ft and an angle of \(0.02094395\) radians, the sector's area calculation was:

• \( A = \frac{1}{2} \times (1.5)^2 \times 0.02094395 \approx 0.0236\) square feet

This area shows how much of the circle is occupied by the sector, emphasizing the practical utility of knowing how to calculate a sector's area. Through this understanding, calculating various measures in circular contexts becomes straightforward.