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Q2(b):
Find the area of the following circles, given that: (b) diameter = $49$ m
Solution :
Given Variables & Initial Setup
We are tasked with determining the area of a two-dimensional circle given its diameter. The primary parameter provided is:
- Diameter ($d$) = $49 \text{ m}$
Step 1: Derivation of the Radius
The radius ($r$) of a circle is defined as the linear distance from the center point to any point on its circumference. [Per Euclidean geometry, the diameter is the longest chord of the circle passing directly through the center, making it exactly twice the length of the radius]. Therefore, we establish the fundamental relationship:
$r = \frac{d}{2}$
Substituting the given diameter into the equation:
$r = \frac{49}{2} \text{ m} = 24.5 \text{ m}$
Geometric Visualization
Below is a scaled geometric representation of the circle, illustrating the relationship between the diameter and the radius.
Step 2: Application of the Area Formula
The area ($A$) of a circle is calculated using the standard formula:
$A = \pi r^2$
[Derived historically via the method of exhaustion by Archimedes, where the area of a circle is proven to be the limit of the areas of inscribed regular polygons as the number of sides approaches infinity].
For this computation, we will utilize the rational approximation $\pi \approx \frac{22}{7}$. This specific approximation is strategically chosen because the radius ($\frac{49}{2}$) contains a numerator that is a multiple of $7$, which will elegantly simplify the subsequent arithmetic.
Step 3: Algebraic Computation
Substitute $r = \frac{49}{2} \text{ m}$ and $\pi = \frac{22}{7}$ into the area formula:
$A = \frac{22}{7} \times \left(\frac{49}{2}\right)^2$
Expand the squared term to prepare for cross-cancellation:
$A = \frac{22}{7} \times \frac{49}{2} \times \frac{49}{2}$
Perform cross-cancellation to simplify the fractional expression. Divide $22$ by $2$ to yield $11$, and divide $49$ by $7$ to yield $7$:
$A = 11 \times 7 \times \frac{49}{2}$
Multiply the integer terms:
$A = 77 \times \frac{49}{2}$
Convert the remaining fraction to a decimal for final multiplication ($ \frac{49}{2} = 24.5 $):
$A = 77 \times 24.5$
Execute the final multiplication:
$A = 1886.5 \text{ m}^2$
Final Solution: The area of the circle is $1886.5 \text{ m}^2$.
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(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 â‚ą $4$ 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 â‚ą $15$. (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 â‚ą $15/m^2$. (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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