Chapter 11 · Areas Related to Circles

NCERT Exemplar Class 10 Maths Exercise 11.2

All 14 questions of NCERT Exemplar Class 10 Maths Exercise 11.2 (Areas Related to Circles), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Is the ​area of the ​circle ​inscribed in ​a square of ​side a cm, $\pi a^2$ cm${}^2$? Give ​reasons for your ​answer.
   Show answer

   False

   Step-by-step solution →
2. Q2
   Will it be true ​to say that ​the perimeter ​of a square circumscribing ​a circle ​of radius $a$ cm is $8a$ cm? Give reasons for your ​answer.
   Show answer

   True. The side of the square equals the diameter $2a$, so the perimeter is $4\times 2a=8a$ cm.

   Step-by-step solution →
3. Q3
   In Fig. ​11.3, a square is inscribed in ​a circle of ​diameter d and another ​square is circumscribing the circle. Is the area of the ​outer square ​four times the ​area of the ​inner square? Give reasons for your answer.

   Show answer

   False

   Step-by-step solution →
4. Q4
   Is it true to say that area ​of a segment of ​a ​circle is less than the ​area ​of its ​corresponding sector? ​Why?
   Show answer

   It is true only for a minor segment (sector = triangle + minor segment, and the triangle has positive area). For a major segment it need not be true, so the statement is not true in general.

   Step-by-step solution →
5. Q5
   Is it true that ​the distance travelled ​by a ​circular wheel ​of diameter d cm ​in ​one ​revolution is $2\pi d$ cm? Why?
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   False

   Step-by-step solution →
6. Q6
   In ​covering a distance $s$ metres, a circular ​wheel of ​radius $r$ ​metres makes $\frac{s}{2\pi r}$ ​revolutions. ​Is this statement true? Why?
   Show answer

   True. One revolution covers $2\pi r$ metres, so covering $s$ metres needs $\frac{s}{2\pi r}$ revolutions.

   Step-by-step solution →
7. Q7
   The ​numerical ​value of the ​area of ​a circle is ​greater ​than ​the numerical value of its circumference. Is this statement true? Why?
   Show answer

   False in general. $A>C$ only when $r>2$; for $r<2$ (e.g. $r=1$) the area is smaller. So the statement is not always true.

   Step-by-step solution →
8. Q8
   If the length of an arc of a circle ​of ​radius r is equal to that of an arc of a circle of radius ​2r, then the angle ​of the corresponding sector ​of the first circle is double the ​angle ​of the corresponding sector of the other circle. Is this statement false? Why?
   Show answer

   False (the statement is correct)

   Step-by-step solution →
9. Q9
   The areas ​of ​two sectors of two different circles with equal corresponding arc ​lengths are equal. Is ​this statement ​true? ​Why?
   Show answer

   False. Sector area = $\frac{1}{2}lr$, so with equal arc length $l$ but unequal radii the areas differ; they are equal only when the radii are equal.

   Step-by-step solution →
10. Q10
    The areas of two ​sectors of two different circles ​are equal. Is ​it necessary ​that their corresponding ​arc lengths ​are equal? ​Why?
    Show answer

    No, it is not necessary. Since $A=\frac{1}{2}r\ell$, equal areas only mean $r_1{\ell }_1=r_2{\ell }_2$; with different radii the arc lengths can differ (e.g. $r=2,\ell =6$ and $r=3,\ell =4$ both give area $6$).

    Step-by-step solution →
11. Q11
    Is ​the ​area of the ​largest ​circle that can be drawn inside a rectangle ​of length $a$ ​cm and breadth $b$ cm $(a>b)$ is $\pi b^2$ cm${}^2$? ​Why?
    Show answer

    No. The largest circle has diameter $b$, so its area is $\pi (b/2{)}^2=\frac{\pi b^2}{4}$ cm${}^2$, not $\pi b^2$ cm${}^2$.

    Step-by-step solution →
12. Q12
    Circumferences of two circles ​are equal. ​Is it necessary that ​their areas ​be ​equal? Why?
    Show answer

    Yes. Equal circumferences force equal radii ($C=2\pi r$), and equal radii give equal areas ($A=\pi r^2$). So their areas must be equal.

    Step-by-step solution →
13. Q13
    Areas of two circles ​are equal. Is it ​necessary ​that their circumferences ​are ​equal? Why?
    Show answer

    Yes, it is necessary. Equal areas $\pi r_1^2=\pi r_2^2$ give $r_1=r_2$, and equal radii give equal circumferences $2\pi r_1=2\pi r_2$.

    Step-by-step solution →
14. Q14
    Is ​it ​true ​to ​say ​that area ​of a square inscribed in a circle of diameter $p$ cm ​is $p^2$ cm${}^2$? ​Why?
    Show answer

    No, it is not true. The diagonal of the inscribed square equals the diameter $p$, so the side is $\frac{p}{\sqrt{2}}$ and the area is $\frac{p^2}{2}$ cm${}^2$, not $p^2$.

    Step-by-step solution →