Question

Find the area of a sector in which the measure of the central angle is \(60^{\circ}\) and the radius of the circle is 2 .

Short answer

The area of the sector is \(\frac{2\pi}{3}\) square units.

Step-by-step solution

Convert the central angle from degrees to radians

To convert the angle from degrees to radians, we use the following formula: \(angle_{radians} = \frac{angle_{degrees} \times \pi}{180}\) Plugging the given angle into the formula, we get: \(angle_{radians} = \frac{60 \times \pi}{180}\)

Simplify the angle in radians

Now, simplify the expression: \(angle_{radians} = \frac{60\pi}{180} = \frac{1\pi}{3}\) So, the central angle is \(\frac{1\pi}{3}\) radians.

Use the formula for the area of a sector

The formula for the area of a sector is: \(area = \frac{1}{2} \times radius^2 \times angle_{radians}\) We know that the radius is 2 and the central angle is \(\frac{1\pi}{3}\) radians. So, plugging these values into the formula, we get: \(area = \frac{1}{2} \times 2^2 \times \frac{1\pi}{3}\)

Simplify the expression for the area

Simplify the expression: \(area = \frac{1}{2} \times 4 \times \frac{1\pi}{3} = 2 \times \frac{1\pi}{3} = \frac{2\pi}{3}\)

Write the answer

The area of the sector is \(\frac{2\pi}{3}\) square units.

Key concepts

Central Angle Conversion

In circle geometry, one important step when dealing with the area of a sector is converting the central angle from degrees to radians. This conversion is crucial because formulas involving circular measurements often use radians rather than degrees. The conversion formula is:

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

To apply this, simply multiply the degree measure of your angle by \( \pi/180 \). For instance, if you have a central angle of \(60^{\circ}\), converting it would look like this:

• \( angle_{radians} = \frac{60 \times \pi}{180} = \frac{\pi}{3} \)

This conversion allows you to utilize the sector area formula effectively, ensuring your calculations conform to the standardized angular measurement system that radians provide.

Radians

Radians offer a more natural measure of angles compared to degrees, especially in mathematical formulas related to circles. In radians, an angle measures the length of the arc that it subtends on a unit circle. A full circle is \(2\pi\) radians, which corresponds to \(360^{\circ}\). This means that:

• \(\pi\) radians = \(180^{\circ}\)
• \(\frac{\pi}{2}\) radians = \(90^{\circ}\)
• \(\frac{\pi}{3}\) radians = \(60^{\circ}\)

Understanding radians is crucial when applying formulas in trigonometry and calculus because most are derived using this measure. For instance, the formula for the area of a sector uses radians to maintain consistency in measurement across mathematical disciplines.

Sector Area Formula

The formula for determining the area of a sector of a circle is straightforward yet powerful. It is expressed as:

• \( area = \frac{1}{2} \times radius^2 \times angle_{radians} \)

This formula derives from the area of an entire circle (\(\pi \times radius^2\)) but proportionally adjusts based on the size of the angle in radians. When calculating the area, ensure:

• Your angle is in radians, not degrees.
• Use the square of the radius as specified.

For example, with a radius of 2 and a central angle of \(\frac{\pi}{3}\), the area calculation is:

• \( area = \frac{1}{2} \times 2^2 \times \frac{\pi}{3} = \frac{2\pi}{3} \)

This equation allows you to find the precise region occupied by the sector within the circle, a frequent necessity in geometric applications.

Circle Geometry

Circle geometry lays the foundation for various mathematical concepts and calculations, such as the area of sectors. A circle is defined by numerous attributes, including its radius, diameter, circumference, and of course, sectors subdividing its area. Key aspects include:

• Radius: The distance from the center to any point on the circle.
• Diameter: Twice the radius.
• Circumference: The total distance around the circle, calculated as \(2\pi\times radius\).
• Sectors: These are portions of the circle defined by a central angle.

When dealing with sectors, understanding the relationship between these elements, especially how angles in both degrees and radians relate to arcs, is vital for accurate calculations. By using circle geometry, one can delve deeper into topics such as arc length, sector area, and segment-based calculations, providing powerful tools for a range of applications.