SSC 10th Geometry Maharashtra Board: Mensuration (Practice Set 7.3)
Question 1
Radius of a circle is $10\text{ cm}$. Measure of an arc of the circle is $54^\circ$. Find the area of the sector associated with the arc. ($\pi = 3.14$)
Solution:
Given:
Radius ($r$) = $10\text{ cm}$
Measure of arc ($\theta$) = $54^\circ$
$\pi = 3.14$To find:
Area of sector ($A$) = ?Formula:
$$A = \frac{\theta}{360} \times \pi r^2$$Calculation:
Substitute the given values into the formula:$$A = \frac{54}{360} \times 3.14 \times (10)^2$$
$$A = \frac{3}{20} \times 3.14 \times 100$$
$$A = 3 \times 3.14 \times 5$$
$$A = 15 \times 3.14 = 47.1\text{ cm}^2$$
Answer:
The area of the sector is $47.1\text{ cm}^2$.
Question 2
Measure of an arc of a circle is $80^\circ$ and its radius is $18\text{ cm}$. Find the length of the arc. ($\pi = 3.14$)
Solution:
Given:
Measure of arc ($\theta$) = $80^\circ$
Radius ($r$) = $18\text{ cm}$
$\pi = 3.14$To find:
Length of arc ($l$) = ?Formula:
$$l = \frac{\theta}{360} \times 2\pi r$$Calculation:
$$l = \frac{80}{360} \times 2 \times 3.14 \times 18$$$$l = \frac{2}{9} \times 2 \times 3.14 \times 18$$
$$l = 4 \times 3.14 \times 2$$
$$l = 8 \times 3.14 = 25.12\text{ cm}$$
Answer:
The length of the arc is $25.12\text{ cm}$.
Question 3
Radius of a sector of a circle is $3.5\text{ cm}$ and length of its arc is $2.2\text{ cm}$. Find the area of the sector.
Solution:
Given:
Radius ($r$) = $3.5\text{ cm}$
Length of arc ($l$) = $2.2\text{ cm}$To find:
Area of sector ($A$) = ?Formula:
$$A = \frac{l \times r}{2}$$Calculation:
$$A = \frac{2.2 \times 3.5}{2}$$$$A = 1.1 \times 3.5 = 3.85\text{ cm}^2$$
Answer:
The area of the sector is $3.85\text{ cm}^2$.
Question 4
Radius of a circle is $10\text{ cm}$. Area of a sector of the circle is $100\text{ cm}^2$. Find the area of its corresponding major sector. ($\pi = 3.14$)
Solution:
Given:
Radius ($r$) = $10\text{ cm}$
Area of minor sector ($A_{\text{minor}}$) = $100\text{ cm}^2$
$\pi = 3.14$To find:
Area of major sector ($A_{\text{major}}$) = ?Formula:
Area of circle = $\pi r^2$
Area of major sector = Area of circle $-$ Area of minor sectorCalculation:
First, calculate the area of the circle:$$\text{Area of circle} = 3.14 \times (10)^2$$
$$\text{Area of circle} = 3.14 \times 100 = 314\text{ cm}^2$$
Now, find the area of the major sector:
$$A_{\text{major}} = 314 - 100 = 214\text{ cm}^2$$
Answer:
The area of the corresponding major sector is $214\text{ cm}^2$.
Question 5
Area of a sector of a circle of radius $15\text{ cm}$ is $30\text{ cm}^2$. Find the length of the arc of the sector.
Solution:
Given:
Radius ($r$) = $15\text{ cm}$
Area of sector ($A$) = $30\text{ cm}^2$To find:
Length of arc ($l$) = ?Formula:
$$A = \frac{l \times r}{2}$$Calculation:
Substitute the given values into the formula:$$30 = \frac{l \times 15}{2}$$
$$60 = 15l$$
$$l = \frac{60}{15} = 4\text{ cm}$$
Answer:
The length of the arc of the sector is $4\text{ cm}$.
