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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 $\pi = 3.14$)
From a circular sheet of radius $4$ cm, a circle of radius $3$ cm is removed. Find the area of the remaining sheet. (Take $\pi = 3.14$)
Solution :
Given Variables & Initial Setup
We are analyzing a composite two-dimensional geometric system where a smaller circle is concentrically removed from a larger circular sheet. The parameters defining this system are:
- Radius of the outer circular sheet, $R = 4 \text{ cm}$
- Radius of the inner removed circle, $r = 3 \text{ cm}$
- Approximation for Pi, $\pi = 3.14$
Geometric Visualization
The figure below illustrates the concentric nature of the two circles. The region between the outer boundary and the inner boundary represents the remaining sheet, geometrically defined as an annulus.
Step 1: Formulating the Area Equation
To find the area of the remaining sheet, we must subtract the area of the inner circle from the area of the outer circular sheet. Let $A$ represent the area of the remaining sheet.
$A = \text{Area of Outer Circle} - \text{Area of Inner Circle}$
$A = \pi R^2 - \pi r^2$ [Per the standard area formula for a circle, $Area = \pi \times \text{radius}^2$]
To optimize the calculation, we factor out the common constant $\pi$:
$A = \pi (R^2 - r^2)$ [By the distributive property of multiplication over subtraction]
Step 2: Algebraic Substitution
Substitute the given numerical values into the factored equation:
$A = 3.14 \times (4^2 - 3^2)$
Step 3: Evaluating the Expression
First, resolve the exponents inside the parentheses:
- $4^2 = 4 \times 4 = 16$
- $3^2 = 3 \times 3 = 9$
Substitute these squared values back into the equation:
$A = 3.14 \times (16 - 9)$
Perform the subtraction operation:
$A = 3.14 \times 7$
Step 4: Final Computation
Multiply the remaining terms to find the final area:
$A = 21.98$
Since the radii were provided in centimeters ($\text{cm}$), the resulting area must be expressed in square centimeters ($\text{cm}^2$) [Per the dimensional analysis of area, $L \times L = L^2$].
Final Solution: The area of the remaining sheet is $21.98 \text{ cm}^2$.
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 $\pi = \frac{22}{7}$)
- 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 $\pi = \frac{22}{7}$)
- 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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