Question

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 \[\pi = \dfrac{{22}}{7}\]

Answer 1

Hint: As we have to bend wire into different shapes, its length will become the perimeter of the respective shapes and the formula for the same can be applied to calculate radius (in case of circle) and side (in case of square).
After finding these values we can substitute these values in the formula of their areas so as to compare which one encloses more area than the other.
Formulas to be used are:
Circumference of circle = $2\pi r$ [r=radius]
Area of circle = $\pi {r^2}$
Perimeter of square = 4a [a=side]
Area of square = ${a^2}$

Complete step-by-step answer:
Length of wire (given) = 44 cm
i) When wire is bent into the shape of a circle:

Length of the wire will be the circumference of circle (as circle can be made out of the available wire length)
Let the radius of the circle be r and it can be calculated as:
Circumference of circle = $2\pi r$
Length of wire = 44 cm.
As these both are equal, we have:
$2\pi r$ = 44 cm
$r = \dfrac{{44}}{{2\pi }}$
Substituting \[\pi = \dfrac{{22}}{7}\] as given
$r = \dfrac{{44 \times 7}}{{2 \times 22}}$
r = 7
Therefore, the radius of the circle formed by the given wire is 7 cm.
Calculating the area of this circle:
Area of circle = $\pi {r^2}$
Substituting the values, we get:
$\pi {r^2}$ = $\dfrac{{22}}{7} \times {\left( 7 \right)^2}$
 = 22 X 7
 = 154
Therefore, the area of the circle formed by the given wire is \[154c{m^2}\] ______ (1)

ii) When wire is bent into the shape of a square:
Length of the wire will be the perimeter of the square (as square can be made out of the available wire length)

Let the side of the square formed be a and it can be calculated as:
Perimeter of square = 4a
Length of the wire = 44 cm
As these both are equal, we have:
4a = 44
a = $\dfrac{{44}}{4}$
a = 11 cm
Therefore, the length of each side of the square formed by the given wire is 11 cm.
Calculating the area of this square:
Area of square = ${a^2}$
Substituting the calculated value of a, we get:
${a^2}$ = ${\left( {11} \right)^2}$
 =121
Therefore, the area of the square formed by the given wire is \[121c{m^2}\] ______ (2)
From (1) and (2), it can be seen that.
Area of circle = \[154c{m^2}\]
Area of square = \[121c{m^2}\]

Thus the circle encloses more area than is enclosed by the square. So, the correct answer is “Option C”.

Note: Perimeter in case of a circle is known as circumference.

For a direct comparison between the area of a circle with diameter $w$ and a square of same width $w$, we can use:

Area of square =side$^2 ={w^2}$

Area of circle = $\pi \text{radius}^2$

$\pi (\dfrac{w}{2})^2$

$\dfrac{\pi }{4} \times {w^2}$