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Q10:
From a circular card sheet of radius $14$ cm, two circles of radius $3.5$ cm and a rectangle of length $3$ cm and breadth $1$cm are removed. (as shown in the adjoining figure). Find the area of the remaining sheet. (Take $\pi = \frac{22}{7}$)
From a circular card sheet of radius $14$ cm, two circles of radius $3.5$ cm and a rectangle of length $3$ cm and breadth $1$cm are removed. (as shown in the adjoining figure). Find the area of the remaining sheet. (Take $\pi = \frac{22}{7}$)
Solution :
Step 1: Given Variables & Initial Setup
To determine the area of the remaining card sheet, we must first establish the dimensions of the original geometric figure and the components that are removed. The problem provides the following parameters:
- Radius of the original large circular sheet, $R = 14 \text{ cm}$
- Radius of the two smaller removed circles, $r = 3.5 \text{ cm}$
- Length of the removed rectangle, $l = 3 \text{ cm}$
- Breadth of the removed rectangle, $b = 1 \text{ cm}$
- Constant for Pi, $\pi = \frac{22}{7}$
Step 2: Visual Representation of the Geometric Configuration
The following scale-accurate diagram illustrates the original circular sheet with the two smaller circles and the rectangle removed. [The diagram uses a scale factor of $10\times$ for visual clarity, where $14 \text{ cm}$ is represented by $140 \text{ units}$].
Step 3: Calculating the Area of the Original Circular Sheet
The area of a circle is given by the formula $A = \pi r^2$. Applying this to the large circular sheet:
$A_{\text{total}} = \pi R^2$
$A_{\text{total}} = \frac{22}{7} \times (14)^2$
$A_{\text{total}} = \frac{22}{7} \times 196$
$A_{\text{total}} = 22 \times 28$
$A_{\text{total}} = 616 \text{ cm}^2$
Step 4: Calculating the Area of the Removed Components
[Per the geometric principle of area addition, we must calculate the individual areas of all removed disjoint regions and sum them].
1. Area of the Two Small Circles:
First, we find the area of one small circle. To simplify the calculation, we can express the radius $3.5 \text{ cm}$ as the fraction $\frac{7}{2} \text{ cm}$.
$A_{\text{small\_circle}} = \pi r^2$
$A_{\text{small\_circle}} = \frac{22}{7} \times \left(\frac{7}{2}\right)^2$
$A_{\text{small\_circle}} = \frac{22}{7} \times \frac{49}{4}$
$A_{\text{small\_circle}} = \frac{22 \times 7}{4} = \frac{154}{4} = 38.5 \text{ cm}^2$
Since two identical circles are removed, their combined area is:
$A_{\text{both\_circles}} = 2 \times 38.5 \text{ cm}^2 = 77 \text{ cm}^2$
2. Area of the Rectangle:
The area of a rectangle is the product of its length and breadth ($A = l \times b$).
$A_{\text{rectangle}} = 3 \text{ cm} \times 1 \text{ cm} = 3 \text{ cm}^2$
3. Total Removed Area:
Summing the areas of the removed components yields:
$A_{\text{removed}} = A_{\text{both\_circles}} + A_{\text{rectangle}}$
$A_{\text{removed}} = 77 \text{ cm}^2 + 3 \text{ cm}^2 = 80 \text{ cm}^2$
Step 5: Calculating the Area of the Remaining Sheet
[By the axiom of area subtraction, the area of a region remaining after removing sub-regions is equal to the area of the original region minus the total area of the removed sub-regions].
$A_{\text{remaining}} = A_{\text{total}} - A_{\text{removed}}$
$A_{\text{remaining}} = 616 \text{ cm}^2 - 80 \text{ cm}^2$
$A_{\text{remaining}} = 536 \text{ cm}^2$
Final Solution: The area of the remaining sheet is $536 \text{ cm}^2$.
More Questions from Class 9 Mathematics Coordinate Geometry EXERCISE 9.2
- Q1(a): Find the circumference of the circles with the following radius: (Take $\pi = \frac{22}{7}$) (a) $14$ cm
- Q1(b): Find the circumference of the circles with the following radius: (Take $\pi = \frac{22}{7}$) (b) $28$ mm
- Q1(c): Find the circumference of the circles with the following radius: (Take $\pi = \frac{22}{7}$) (c) $21$ cm
- Q11: A circle of radius $2$ cm is cut out from a square piece of an aluminium sheet of side $6$ cm. What is the area of the left over aluminium sheet? (Take $\pi = 3.14$)
- Q12: The circumference of a circle is $31.4$ cm. Find the radius and the area of the circle? (Take $\pi = 3.14$)
- Q13: A circular flower bed is surrounded by a path $4$ m wide. The diameter of the flower bed is $66$ m. What is the area of this path? ($\pi = 3.14$)
- Q14: A circular flower garden has an area of $314$ m$^2$. A sprinkler at the centre of the garden can cover an area that has a radius of $12$ m. Will the sprinkler water the entire garden? (Take $\pi = 3.14$)
- Q15: Find the circumference of the inner and the outer circles, shown in the adjoining figure? (Take $\pi = 3.14$)
- Q16: How many times a wheel of radius $28$ cm must rotate to go $352$ m? (Take $\pi = \frac{22}{7}$)
- Q17: The minute hand of a circular clock is $15$ cm long. How far does the tip of the minute hand move in $1$ hour. (Take $\pi = 3.14$)
- Q2(a): Find the area of the following circles, given that: (a) radius = $14$ mm (Take $\pi = \frac{22}{7}$)
- Q2(b): Find the area of the following circles, given that: (b) diameter = $49$ m
- Q2(c): Find the area of the following circles, given that: (c) radius = $5$ cm
- Q3: If the circumference of a circular sheet is $154$ m, find its radius. Also find the area of the sheet. (Take $\pi = \frac{22}{7}$)
- Q4: A gardener wants to fence a circular garden of diameter $21$ m. Find the length of the rope he needs to purchase, if he makes $2$ rounds of fence. Also find the cost of the rope, if it costs ₹ $4$ per meter. (Take $\pi = \frac{22}{7}$)
- Q5: From a circular sheet of radius $4$ cm, a circle of radius $3$ cm is removed. Find the area of the remaining sheet. (Take $\pi = 3.14$)
- Q6: Saima wants to put a lace on the edge of a circular table cover of diameter $1.5$ m. Find the length of the lace required and also find its cost if one meter of the lace costs ₹ $15$. (Take $\pi = 3.14$)
- Q7: Find the perimeter of the adjoining figure, which is a semicircle including its diameter.
- Q8: Find the cost of polishing a circular table-top of diameter $1.6$ m, if the rate of polishing is ₹ $15/m^2$. (Take $\pi = 3.14$)
- Q9: Shazli took a wire of length $44$ cm and bent it into the shape of a circle. Find the radius of that circle. Also find its area. If the same wire is bent into the shape of a square, what will be the length of each of its sides? Which figure encloses more area, the circle or the square? (Take $\pi = \frac{22}{7}$)
CBSE Solutions for Class 9 Mathematics Coordinate Geometry
Chapters in CBSE - Class 9 Mathematics
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