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Q1(b):
Find the circumference of the circles with the following radius: (Take $\pi = \frac{22}{7}$) (b) $28$ mm
Solution :
Given Variables & Initial Setup
We are tasked with determining the circumference of a circle given its radius. The fundamental parameters provided for this geometric system are:
- Radius of the circle, $r = 28 \text{ mm}$
- Approximation for Pi, $\pi = \frac{22}{7}$
Theoretical Foundation: Circumference of a Circle
The circumference ($C$) of a circle is defined as the continuous line forming the boundary of the closed geometric figure. [Per the geometric definition of a circle's perimeter in Euclidean space], the circumference is directly proportional to its radius, with the constant of proportionality being $2\pi$. The governing formula is:
$C = 2\pi r$
Step 1: Substitution of Values
We substitute the given values of $r$ and $\pi$ into the standard circumference formula:
$C = 2 \times \left(\frac{22}{7}\right) \times 28$
Step 2: Algebraic Simplification
To optimize the calculation, we first analyze the terms for common factors. The radius ($28$) is a multiple of the denominator ($7$) in our fractional value of $\pi$. [By the fundamental properties of rational numbers], we can divide $28$ by $7$ to simplify the expression before proceeding with multiplication:
$\frac{28}{7} = 4$
Substituting this simplified integer back into the equation yields:
$C = 2 \times 22 \times 4$
Step 3: Final Calculation
We now perform the sequential multiplication of the remaining scalar quantities:
First, multiply the constants:
$2 \times 22 = 44$
Next, multiply the result by the simplified radius factor:
$C = 44 \times 4$
$C = 176$
Since the radius was provided in millimeters ($\text{mm}$), the circumference, being a one-dimensional measure of length, must carry the identical unit.
Final Solution: The circumference of the circle is $176 \text{ mm}$.
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(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 $\pi = \frac{22}{7}$)
- 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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