Solving the Circular Sector Area Problem

This problem asks us to find the area of a circular sector given its perimeter and radius. We are provided with the radius and the total length of the boundary of the sector (the perimeter). To find the area, we first need to determine the central angle of the sector.

Understanding Circular Sectors and Formulas

A circular sector is a portion of a disk (a circle) enclosed by two radii and the intercepted arc. Its perimeter consists of the two radii and the arc length. Its area is a fraction of the area of the whole circle, determined by the central angle.

The key formulas for a circular sector are:

• Perimeter of a sector = 2 × radius + arc length
• Arc length of a sector = \(r \theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.
• Area of a sector = \(\frac{1}{2} r^2 \theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians. Alternatively, Area = \(\frac{1}{2} \times \text{arc length} \times r\).

Step-by-Step Calculation

We are given:

• Radius (\(r\)) = 5.2 cm
• Perimeter of the sector = 16.4 cm

Step 1: Find the Arc Length

The perimeter of the sector is the sum of the lengths of the two radii and the arc.

Perimeter = \(2r\) + Arc length

We can plug in the given values:

\(16.4 = 2 \times 5.2\) + Arc length

\(16.4 = 10.4\) + Arc length

Subtract 10.4 from both sides to find the arc length:

Arc length = \(16.4 - 10.4\)

Arc length = 6 cm

Step 2: Calculate the Area using Arc Length

We can use the formula for the area of a sector that involves the arc length and the radius:

Area = \(\frac{1}{2} \times \text{arc length} \times r\)

Now, substitute the values we found:

Area = \(\frac{1}{2} \times 6 \times 5.2\)

Area = \(3 \times 5.2\)

Area = 15.6 square cm

Alternatively, we could first find the angle \(\theta\) in radians using the arc length formula (\(\text{Arc length} = r \theta\)):

\(6 = 5.2 \times \theta\)

\(\theta = \frac{6}{5.2}\) radians

Then use the area formula \(Area = \frac{1}{2} r^2 \theta\):

Area = \(\frac{1}{2} \times (5.2)^2 \times \frac{6}{5.2}\)

Area = \(\frac{1}{2} \times 5.2 \times 5.2 \times \frac{6}{5.2}\)

Area = \(\frac{1}{2} \times 5.2 \times 6\)

Area = \(2.6 \times 6\)

Area = 15.6 square cm

Both methods give the same result for the area of the circular sector.

Revision Table: Circular Sector Formulas

Additional Information: Understanding Radians

In the formulas for arc length and area of a sector (\(L = r \theta\) and \(A = \frac{1}{2} r^2 \theta\)), the angle \(\theta\) must be in radians. A radian is a unit of angular measurement. One radian is defined as the angle subtended at the center of a circle by an arc that is equal in length to the radius.

• Conversion between degrees and radians:
• \(\pi\) radians = 180°
• 1 radian = \(\frac{180}{\pi}\) degrees
• 1 degree = \(\frac{\pi}{180}\) radians

Using radians simplifies many formulas in calculus and geometry involving circles and circular motion, including the sector area and arc length formulas used in this problem.