Question

Find the area of a \(30^{\circ}\) sector of a circle if the radius of the circle is 36 meters. Round to the nearest hundredth. (Lesson 11-6)

Short answer

The area of the sector is approximately 339.29 square meters.

Step-by-step solution

Understanding the Sector Area Formula

To find the area of a sector, we use the formula \( \text{Area of Sector} = \frac{\theta}{360^{\circ}} \times \pi r^2 \), where \( \theta \) is the central angle of the sector in degrees, and \( r \) is the radius of the circle.

Substituting Known Values

Given \( \theta = 30^{\circ} \) and \( r = 36 \) meters, substitute these into the formula: \( \text{Area of Sector} = \frac{30}{360} \times \pi \times 36^2 \).

Simplifying the Fraction

The fraction simplifies to \( \frac{30}{360} = \frac{1}{12} \). This reduces the formula to \( \text{Area of Sector} = \frac{1}{12} \times \pi \times 36^2 \).

Calculating the Radius Squared

Calculate \( 36^2 = 1296 \), so the next step will be to substitute this into the formula, resulting in \( \text{Area of Sector} = \frac{1}{12} \times \pi \times 1296 \).

Multiplying by \( \pi \) and Simplification

Now calculate \( 1296 \times \pi \). Solve \( \text{Area of Sector} = \frac{1}{12} \times 1296\pi \), which is \( 108\pi \) when multiplied by \( \frac{1}{12} \).

Converting \( \pi \) into Decimal and Final Calculation

Using \( \pi \approx 3.1416 \) (rounded to four decimal places), we find that \( 108\pi \approx 339.2928 \).

Rounding the Final Answer

Round \( 339.2928 \) to the nearest hundredth to get \( 339.29 \). Thus, the area of the sector is approximately 339.29 square meters.

Key concepts

Circle Geometry

Circle geometry is a fascinating topic in mathematics that delves into the properties and relationships between different components of a circle.
A circle is a perfect closed curve where every point on the curve is equidistant from a fixed point called the center.
In circle geometry, we explore various elements like the radius, diameter, and circumference, but one interesting part is the sector, which is a "slice" of the circle.

• The radius is the distance from the center to any point on the circle.
• The diameter, which is twice the radius, spans across the circle through the center.
• The circumference is the total distance around the circle.

A sector is formed by two radii and the arc between them. It's shaped like a pizza slice, and to find its area, we use properties of the circle along with the central angle, which measures the openness of the slice.

Central Angle

The central angle is a crucial component in determining the sector area. It is the angle with its vertex at the circle's center and its sides being radii that intersect the circle in two places.
Depending on its measurement, it gives us an idea of how large the sector is in relation to the entire circle.

• A full circle is comprised of 360 degrees, hence a central angle divides the circle into specific portions or sectors.
• In our problem, a 30-degree angle forms a sector, meaning this sector covers just 30 out of 360 parts of the circle.

This ratio, \frac{\theta}{360^{\circ}}\, where \( \theta \) is the central angle, is pivotal in calculating the actual area of the sector compared to the whole circle.

Radians and Degrees

When discussing circles and angles, understanding radians and degrees is vital. These are two units of angular measurement that allow us to translate concepts from circular geometry into calculations.

• Degrees divide a circle into 360 equally spaced increments.
• Radians provide an alternative measurement, where one radian equals the angle formed when the arc length equals the radius of the circle.

One full circle constitutes \( 2\pi \) radians, equating \( 360^{\circ} \). Thus, to convert degrees to radians, you can use the formula: \[ \ heta_{radians} = \ heta_{degrees} \times \frac{\pi}{180^{\circ}} \].
This conversion is especially useful in advanced calculus and trigonometry but is not necessarily required for simpler sector area calculations, which can be done directly in degrees.

Mathematical Formula

To find the area of a sector, we utilize a simple yet powerful mathematical formula derived from the circle's principles.
The formula: \[ \text{Area of Sector} = \frac{\theta}{360^{\circ}} \times \pi r^2 \] allows us to calculate the area of any sector, provided we have the central angle in degrees and the circle's radius.
This formula works because:

• \( \frac{\theta}{360^{\circ}} \) calculates the fraction of the circle which the sector represents.
• \( \pi r^2 \) is the formula for the area of the entire circle.

By multiplying these two elements, we find the specific area occupied by the sector. This formula is thus fundamental in any mathematical problem related to circle geometry, such as the textbook exercise discussed.