Problem 92 - Precalculus

Question

Find the area of the circular sector given the indicated radius and central angle. Round your answers to three significant digits. $$\theta=27^{\circ}, r=2.5 \mathrm{mm}$$

Step-by-Step Solution

Answer

The area of the circular sector is approximately 1.47 mm².

Step 1: Convert Degrees to Radians

The formula to convert degrees to radians is \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \). Given \( \theta = 27^{\circ} \), we have: \( \theta_{radians} = 27 \times \frac{\pi}{180} = \frac{27\pi}{180} = \frac{3\pi}{20} \approx 0.4712 \text{ radians} \) after rounding.

Step 2: Use Sector Area Formula

The area \( A \) of a 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 \( r = 2.5 \) mm and \( \theta = 0.4712 \) radians, we calculate: \( A = \frac{1}{2} (2.5)^2 (0.4712) \).

Step 3: Calculate the Area

Perform the calculations: \( A = \frac{1}{2} \times 6.25 \times 0.4712 = 1.4735 \text{ mm}^2 \).

Step 4: Round to Three Significant Digits

Round the calculated area to three significant digits to get \( A \approx 1.47 \text{ mm}^2 \).

Key Concepts

Degrees to Radians Conversion

To understand how to convert degrees to radians, think of the mathematical relationship between these two units of angle measurements. Degrees and radians are two different ways to describe angles. However, in many mathematical formulas, radians are preferred.
To convert degrees into radians, use the formula:

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

For instance, if you have an angle of \(27^{\circ}\), you multiply it by \(\frac{\pi}{180}\) to convert it into radians:

• \( \theta_{radians} = 27 \times \frac{\pi}{180} = \frac{3\pi}{20} \approx 0.4712 \text{ radians}\)

This conversion is essential because the sectors' formulas often use radians.

Sector Area Formula

When you want to find the area of a sector, which is like a 'slice' of a circle, you use the sector area formula. This formula helps you calculate how much space that sector occupies.
The sector area formula is:

• \( A = \frac{1}{2} r^2 \theta \)

This means you take half of the product of the radius squared and the central angle in radians. Let's break this down with our previous example:

• Given \( r = 2.5 \) mm and \( \theta = 0.4712 \) radians, substitute these values into the formula.
• \( A = \frac{1}{2} (2.5)^2 (0.4712) = \frac{1}{2} \times 6.25 \times 0.4712 \)

After calculating, you find the area to be approximately \(1.4735 \text{ mm}^2\).

Central Angle

The central angle is the angle originating from the center of the circle, subtending an arc of the circle. It is what defines the size of a sector.
When calculating the area of a circular sector, it's necessary to express the central angle in radians.
In our exercise, we started with a central angle of \(27^{\circ}\) which was converted into radians.
This step is crucial because in mathematical calculations involving arc length or sector area, the radian measure is used because it provides a direct correlation with the arc length and radius.

Radius

The radius is a key factor in the formula for calculating the area of a sector. It is the distance from the center of the circle to any point on its circumference.
In our example, the radius is \(2.5 \text{ mm}\). The radius determines how large the circle itself is, and consequently, how large each sector of that circle can be.
Using the sector area formula, the radius is squared, which means small changes in the radius can lead to significant changes in the area calculation.
Remember, it's important to ensure the radius is correctly measured or given as it directly influences the resulting area of the sector.