Sector Perimeter Calculation for Circle

This explanation details how to find the perimeter of a sector in a circle, given its radius and angle. We will use the provided values and standard geometry formulas to solve the problem.

Given Information Analysis

• Radius ($r$): The radius of the circle is provided as $10.5$ cm.
• Sector Angle ($\theta$): The angle subtended by the sector at the center is $\frac{2\pi}{3}$ radians.
• Value of Pi ($\pi$): We are instructed to use the approximation $\pi = \frac{22}{7}$ for calculations.

Formula for Sector Perimeter

The perimeter of a circular sector is the sum of the lengths of the two radii and the arc length along the boundary. The formula is:

Perimeter = $2 \times \text{Radius} + \text{Arc Length}$

The arc length ($L$) of a sector can be calculated using the formula:

Arc Length ($L$) = $r \theta$, where $\theta$ must be in radians.

Step-by-Step Calculation

1. Calculate the Arc Length:

   First, we find the length of the arc using the given radius and angle.

   Arc Length = $r \theta = 10.5 \text{ cm} \times \frac{2\pi}{3}$

   To make the calculation easier, let's convert the radius $10.5$ cm into a fraction: $10.5 = \frac{21}{2}$.

   Now, substitute the value of $\pi = \frac{22}{7}$ into the arc length formula:

   Arc Length = $\frac{21}{2} \times \frac{2}{3} \times \frac{22}{7}$ cm

   Perform the multiplication:

   Arc Length = $\frac{21 \times 2 \times 22}{2 \times 3 \times 7}$ cm

   Simplify the expression by cancelling common factors. Notice that $21 = 3 \times 7$.

   Arc Length = $\frac{(3 \times 7) \times 2 \times 22}{2 \times 3 \times 7}$ cm

   After cancellation, we get:

   Arc Length = $22$ cm

2. Calculate the Perimeter of the Sector:

   Next, we use the arc length we just calculated and the given radius to find the total perimeter.

   Perimeter = $2r + \text{Arc Length}$

   Substitute the values: $r = 10.5$ cm and Arc Length = $22$ cm.

   Perimeter = $(2 \times 10.5 \text{ cm}) + 22 \text{ cm}$

   Calculate the length of the two radii:

   Perimeter = $21 \text{ cm} + 22 \text{ cm}$

   Add the values to find the total perimeter:

   Perimeter = $43$ cm

Final Result

Therefore, the perimeter of the sector is 43 cm.