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Q2(a):
Find the area of the following circles, given that: (a) radius = $14$ mm (Take $\pi = \frac{22}{7}$)
Solution :
Given Variables & Initial Setup
We are given the following geometric parameters for a circle:
- Radius ($r$) = $14 \text{ mm}$
- Mathematical constant ($\pi$) = $\frac{22}{7}$
Theoretical Foundation & Formula
The area ($A$) of a circle represents the total two-dimensional space enclosed within its boundary (circumference). [Per the geometric principles of Euclidean space], the area of a circle is directly proportional to the square of its radius. The formula is given by:
$A = \pi r^2$
Step 1: Substituting the Values into the Equation
To find the area, we substitute the given radius $r = 14 \text{ mm}$ and the fractional approximation of $\pi = \frac{22}{7}$ into the standard area formula:
$A = \left(\frac{22}{7}\right) \times (14 \text{ mm})^2$
Step 2: Algebraic Manipulation and Simplification
First, expand the squared term to separate the numerical values from the units:
$A = \frac{22}{7} \times (14 \times 14) \text{ mm}^2$
$A = \frac{22}{7} \times 196 \text{ mm}^2$
To optimize the calculation without dealing with large numbers, we can simplify the expression by dividing one of the $14$ factors by the denominator $7$ [By the fundamental property of fractions and associative multiplication]:
$A = 22 \times \left(\frac{14}{7}\right) \times 14 \text{ mm}^2$
$A = 22 \times 2 \times 14 \text{ mm}^2$
Now, proceed with sequential multiplication:
$A = (22 \times 2) \times 14 \text{ mm}^2$
$A = 44 \times 14 \text{ mm}^2$
Perform the final multiplication step:
$A = 44 \times (10 + 4) \text{ mm}^2$
$A = 440 + 176 \text{ mm}^2$
$A = 616 \text{ mm}^2$
Step 3: Dimensional Analysis
The radius is given in millimeters ($\text{mm}$). When the radius is squared in the formula ($r^2$), the unit is also squared ($\text{mm} \times \text{mm} = \text{mm}^2$). This confirms that our final unit correctly represents a two-dimensional area.
Final Solution: The area of the circle is $616 \text{ mm}^2$.
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(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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