Mensuration: Area and Perimeter - Derive and apply area formulae for circles and sectors

Find the area and circumference of a circle with radius $r = 7$ cm. (Take $\pi = \frac{22}{7}$)

$C = 2 \times \frac{22}{7} \times 7 = 44\text{ cm}$ $A = \frac{22}{7} \times 7 \times 7 = 154\text{ cm}^2$

Substitute the value of $r = 7$ into the formulae for circumference $2\pi r$ and area $\pi r^2$.

Calculate the area of a sector of a circle with radius $21$ cm and a central angle of $60^\circ$.

$Area = \frac{60^\circ}{360^\circ} \times \frac{22}{7} \times 21 \times 21$ $Area = \frac{1}{6} \times 22 \times 3 \times 21$ $Area = 11 \times 21 = 231\text{ cm}^2$

The area is found using the formula $\frac{\theta}{360^\circ} \times \pi r^2$, where $\theta = 60^\circ$ and $r = 21$ cm.

If the circumference of a circular sheet is $154$ m, find its radius and area. (Take $\pi = \frac{22}{7}$)

First, find $r$: $2 \times \frac{22}{7} \times r = 154$ $r = \frac{154 \times 7}{44} = \frac{7 \times 7}{2} = 24.5\text{ m}$ Now, find Area $A$: $A = \frac{22}{7} \times \frac{49}{2} \times \frac{49}{2}$ $A = 11 \times 7 \times \frac{49}{2} = \frac{3773}{2} = 1886.5\text{ m}^2$

First, use the circumference formula $2\pi r = 154$ to solve for $r$. Then, use $r$ to calculate the area using $\pi r^2$.

Subtract the area of a small circle of radius $3$ cm from a large circle of radius $10$ cm. (Use $\pi = 3.14$)

Area of large circle $A_1 = 3.14 \times 10^2 = 314\text{ cm}^2$. Area of small circle $A_2 = 3.14 \times 3^2 = 3.14 \times 9 = 28.26\text{ cm}^2$. Difference: $\begin{array}{r} 314.00 \\ - 28.26 \\ \hline 285.74 \end{array}$ Result = $285.74\text{ cm}^2$.

Calculate the area of both circles separately and subtract the smaller area from the larger area to find the remaining region (annulus).

$Area\ of\ Track = Area\ of\ Outer\ Circle - Area\ of\ Inner\ Circle$ $Area = \pi R^2 - \pi r^2 = \pi(R^2 - r^2)$ $Area = \frac{22}{7} \times (21^2 - 14^2)$ $Area = \frac{22}{7} \times (441 - 196)$ $Area = \frac{22}{7} \times 245 = 22 \times 35 = 770\text{ m}^2$

To find the area of a ring (annulus), we subtract the area of the smaller inner circle from the larger outer circle.

$Perimeter = \text{Arc Length of Semicircle} + \text{Diameter}$ $Arc\ Length = \frac{1}{2} \times 2\pi r = \pi r$ $Arc\ Length = \frac{22}{7} \times 7 = 22\text{ cm}$ $Diameter = 2r = 2 \times 7 = 14\text{ cm}$ $Total\ Perimeter = 22 + 14 = 36\text{ cm}$

The perimeter of a closed semi-circle is the sum of the curved boundary (half the circumference) and the straight boundary (the diameter).