Mensuration - Arc Length and Sector Area

A sector of a circle has a radius of $10\text{ cm}$ and a central angle of $72^\circ$. Calculate the length of the arc. (Take $\pi = 3.142$)

$l = \frac{72}{360} \times 2 \times 3.142 \times 10 = 0.2 \times 62.84 = 12.568\text{ cm}$

Identify the fraction of the circle by dividing the angle $72^\circ$ by $360^\circ$. Multiply this fraction by the full circumference ($2\pi r$) to find the arc length.

Find the area of a sector with a radius of $6\text{ cm}$ and a central angle of $120^\circ$. Give your answer in terms of $\pi$.

$A = \frac{120}{360} \times \pi \times 6^2 = \frac{1}{3} \times 36\pi = 12\pi\text{ cm}^2$

Use the sector area formula. Since $120/360$ simplifies to $1/3$, the area is exactly one-third of the total area of the circle ($36\pi$).

The area of a sector is $25\pi\text{ cm}^2$ and its radius is $10\text{ cm}$. Find the central angle $\theta$.

$25\pi = \frac{\theta}{360} \times \pi \times 10^2 \Rightarrow 25\pi = \frac{\theta \times 100\pi}{360} \Rightarrow 25 = \frac{10\theta}{36} \Rightarrow \theta = \frac{25 \times 36}{10} = 90^\circ$

Substitute the known values (Area and Radius) into the sector area formula and solve for the unknown angle $\theta$ by rearranging the equation.

Calculate the total perimeter of a sector with radius $7\text{ cm}$ and central angle $90^\circ$. (Use $\pi = \frac{22}{7}$)

Arc Length $= \frac{90}{360} \times 2 \times \frac{22}{7} \times 7 = \frac{1}{4} \times 44 = 11\text{ cm}$. Total Perimeter $= 11 + 7 + 7 = 25\text{ cm}$.

First, calculate the arc length. Then, remember that the perimeter of a sector consists of the arc length plus two radii ($r + r$).

$\text{Arc Length} = \frac{45}{360} \times 2 \times \frac{22}{7} \times 14$ $\text{Arc Length} = \frac{1}{8} \times 2 \times 22 \times 2$ $\text{Arc Length} = \frac{1}{8} \times 88 = 11\text{ cm}$ $\text{Perimeter} = \text{Arc Length} + 2r$ $\text{Perimeter} = 11 + 2(14)$ $\text{Perimeter} = 11 + 28 = 39\text{ cm}$

First, calculate the arc length using the formula for a $45^\circ$ angle. Then, add the two radii to the arc length to find the total perimeter of the shape.

$\text{Major Angle} = 360^\circ - 60^\circ = 300^\circ$ $\text{Area} = \frac{300}{360} \times \pi \times 9^2$ $\text{Area} = \frac{5}{6} \times \pi \times 81$ $\text{Area} = \frac{5 \times 27}{2} \pi$ $\text{Area} = 67.5\pi\text{ cm}^2$

To find the area of the major sector, first subtract the given minor angle from $360^\circ$ to find the major central angle. Then apply the sector area formula.