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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 $\pi = 3.14$)
Solution :
Step 1: Given Variables & Initial Setup
To determine the area of the remaining aluminium sheet, we must first establish the dimensions of the primary geometric figures involved in the system. Let $s$ represent the side length of the square aluminium sheet, and $r$ represent the radius of the circular cutout.
- Side of the square sheet, $s = 6 \text{ cm}$
- Radius of the circular cutout, $r = 2 \text{ cm}$
- Constant approximation, $\pi = 3.14$
Step 2: Geometric Visualization
The following scale diagram illustrates the spatial relationship between the square sheet and the circular cutout. The dimensions are mapped to a precise coordinate system where $1 \text{ cm} = 40 \text{ units}$.
Step 3: Calculating the Area of the Square Sheet
The total area of the initial aluminium sheet is calculated using the standard area formula for a regular quadrilateral [Per Euclidean Geometry principles].
$A_{\text{square}} = s^2$
$A_{\text{square}} = (6 \text{ cm})^2$
$A_{\text{square}} = 36 \text{ cm}^2$
Step 4: Calculating the Area of the Circular Cutout
The area of the removed material is defined by the area of the circle. We apply the area formula for a circle, substituting the given approximation for $\pi$.
$A_{\text{circle}} = \pi r^2$
$A_{\text{circle}} = 3.14 \times (2 \text{ cm})^2$
$A_{\text{circle}} = 3.14 \times 4 \text{ cm}^2$
$A_{\text{circle}} = 12.56 \text{ cm}^2$
Step 5: Determining the Area of the Leftover Sheet
To find the area of the remaining aluminium, we subtract the area of the circular cutout from the total area of the square sheet [By the Additive Property of Area, which states that the area of a whole is equal to the sum of the areas of its non-overlapping parts].
$A_{\text{leftover}} = A_{\text{square}} - A_{\text{circle}}$
$A_{\text{leftover}} = 36 \text{ cm}^2 - 12.56 \text{ cm}^2$
$A_{\text{leftover}} = 23.44 \text{ cm}^2$
Summary of Area Calculations
| Geometric Component | Mathematical Formula | Calculated Area |
|---|---|---|
| Original Square Sheet | $A = s^2$ | $36.00 \text{ cm}^2$ |
| Circular Cutout | $A = \pi r^2$ | $12.56 \text{ cm}^2$ |
| Remaining Sheet | $A_{\text{square}} - A_{\text{circle}}$ | $23.44 \text{ cm}^2$ |
Final Solution: The area of the left over aluminium sheet is $23.44 \text{ cm}^2$.
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
- 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 $\pi = \frac{22}{7}$)
- 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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