(Take $\pi = \frac{22}{7}$)
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
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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
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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.
Important Questions from 2-D Mensuration
- The length of a rectangular plot is $(x^2 + xy + y^2)$ m and its breadth is $(x^2 - 5xy - y^2)$ m.
Find its perimeter, when $x = 1$ and $y = -1$. - The ratio between the perimeter and breadth of a rectangle is 3: 1. If the area of the rectangle is $98 \text{ cm}^2$, find the perimeter (in cm) of the rectangle.
The shorter side of a rectangle is 15 cm less than the longer side. The numerical value of its area is equal to 5 times the numerical value of its perimeter. What is the length (in cm) of its longer side?
- Find the circumference (in m) of the largest circle that can be inscribed in a rectangle whose dimensions are given as 114 m and 63 m.
Take $\pi = \frac{22}{7}$ - The length of a rectangular pitch is 30 m more than its breadth. Its area is $18,271$ m$^{2}$. Its breadth (in m) is:
