Mensuration: Area and Perimeter - Compute arc length and apply to circular path and sector problems

Find the length of the arc and the area of a sector of a circle with radius $r = 21\text{ cm}$ and a central angle $\theta = 60^\circ$. (Take $\pi = \frac{22}{7}$)

1. Length of arc $l = \frac{\theta}{360^\circ} \times 2\pi r$ $l = \frac{60}{360} \times 2 \times \frac{22}{7} \times 21$ $l = \frac{1}{6} \times 2 \times 22 \times 3 = 22\text{ cm}$.

2. Area of sector $A = \frac{\theta}{360^\circ} \times \pi r^2$ $A = \frac{60}{360} \times \frac{22}{7} \times 21 \times 21$ $A = \frac{1}{6} \times 22 \times 3 \times 21 = 231\text{ cm}^2$.

We use the central angle ratio $\frac{60}{360}$ and multiply by the circumference formula for arc length and the area formula for the sector area.

A bicycle wheel of radius $35\text{ cm}$ is making revolutions to cover a distance of $110\text{ m}$. Calculate the number of revolutions.

1. Distance in cm: $110\text{ m} = 11000\text{ cm}$.
2. Circumference $C = 2\pi r = 2 \times \frac{22}{7} \times 35 = 220\text{ cm}$.
3. Number of revolutions $n = \frac{\text{Total Distance}}{\text{Circumference}}$: $\begin{array}{r} 11000 \div 220 \\ \hline 50 \end{array}$ So, $n = 50$.

The distance covered is the product of the number of revolutions and the circumference. We converted the units to be consistent (cm) before dividing.

Find the perimeter of a sector of a circle with radius $10.5\text{ cm}$ and central angle $90^\circ$.

1. Arc length $l = \frac{90}{360} \times 2 \times \frac{22}{7} \times 10.5$ $l = \frac{1}{4} \times 2 \times \frac{22}{7} \times \frac{21}{2} = \frac{1}{4} \times 22 \times 3 = 16.5\text{ cm}$.
2. Perimeter $P = l + 2r$ $P = 16.5 + 2(10.5)$ $\begin{array}{r} 16.5 \\ + 21.0 \\ \hline 37.5 \end{array}$ $P = 37.5\text{ cm}$.

The perimeter of a sector includes the curved arc and the two straight radii. First, calculate the arc length, then add twice the radius.

$\text{Radius } r = \frac{\text{diameter}}{2} = \frac{56}{2} = 28\text{ cm}$ $\text{Distance to cover} = 880\text{ m} = 88000\text{ cm}$ $\text{Circumference } C = 2 \pi r = 2 \times \frac{22}{7} \times 28 = 2 \times 22 \times 4 = 176\text{ cm}$ $\text{Number of revolutions } n = \frac{\text{Total Distance}}{\text{Circumference}}$ $\begin{array}{r} 88000 / 176 \\ \hline 500 \end{array}$ $n = 500$

First, find the radius and convert the total distance into the same units (cm). Then calculate the distance covered in one revolution (the circumference). Divide the total distance by the circumference to find the number of revolutions.

$\text{Given: } r = 14\text{ cm}, l = 22\text{ cm}$ $\text{Area of sector } A = \frac{1}{2} \times l \times r$ $A = \frac{1}{2} \times 22 \times 14$ $A = 11 \times 14$ $A = 154\text{ cm}^2$

When the arc length and radius are known, the simplest way to calculate the sector area is using the formula $A = \frac{1}{2}lr$, which is derived from the relationship between arc length and circumference.