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Mathematics · Textbook solutions

Angle and its Measurement

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 99 questions

1.1 Directed Angles

14 q

Solved Examples

Worked · 14
  1. Convert the following degree measures in the radian measures.
    1.1 SolvedEx.1(i)
    70∘70^\circ
  2. 1.1 SolvedEx.1(ii)
    −120∘-120^\circ
  3. 1.1 SolvedEx.1(iii)
    (14)∘\left(\frac{1}{4}\right)^\circ
  4. Convert the following radian measures in the degree measures.
    1.1 SolvedEx.2(i)
    (7π3)c\left(\frac{7\pi}{3}\right)^{c}
  5. 1.1 SolvedEx.2(ii)
    (−π18)c\left(\frac{-\pi}{18}\right)^{c}
  6. 1.1 SolvedEx.2(iii)
    (47)c\left(\frac{4}{7}\right)^{c}
  7. Express the following angles in degree, minute and second.
    1.1 SolvedEx.3(i)
    74.87∘74.87^\circ
  8. 1.1 SolvedEx.3(ii)
    −30.6947∘-30.6947^\circ
  9. 1.1 SolvedEx.4
    The measures of the angles of the triangle are in A. P. The smallest angle is 40. Find the angles of the triangle in degree and in radian.
  10. 1.1 SolvedEx.5
    The difference between two acute angles of a right angled triangle is 7πc30\frac{7\pi^{c}}{30}. Find the angles of the triangle in degree.
  11. 1.1 SolvedEx.6
    One angle of a quadrilateral is 2π9\frac{2\pi}{9} radian and the measures of the other three angles are in the ratio 3:5:8, find their measures in degree.
  12. 1.1 SolvedEx.7
    Find the number of sides of a regular polygon if each of its interior angle is (4π5)c\left(\frac{4\pi}{5}\right)^{c}.
  13. Find the angle between hour hand and minute hand of a clock at
    1.1 SolvedEx.8(i)
    Quarter past five
  14. 1.1 SolvedEx.8(ii)
    Quarter to twelve

Exercise 1.1

48 q
  1. Determine which of the following pairs of angles are co-terminal.
    Ex 1.1 Q1A(i)
    210∘, −150∘210^\circ,\ -150^\circ
  2. Ex 1.1 Q1A(ii)
    360∘, −30∘360^\circ,\ -30^\circ
  3. Ex 1.1 Q1A(iii)
    −180∘, 540∘-180^\circ,\ 540^\circ
  4. Ex 1.1 Q1A(iv)
    −405∘, 675∘-405^\circ,\ 675^\circ
  5. Ex 1.1 Q1A(v)
    860∘, 580∘860^\circ,\ 580^\circ
  6. Ex 1.1 Q1A(vi)
    900∘, −900∘900^\circ,\ -900^\circ
  7. Draw the angles of the following measures and determine their quadrants.
    Ex 1.1 Q1B(i)
    −140∘-140^\circ
  8. Ex 1.1 Q1B(ii)
    250∘250^\circ
  9. Ex 1.1 Q1B(iii)
    420∘420^\circ
  10. Ex 1.1 Q1B(iv)
    750∘750^\circ
  11. Ex 1.1 Q1B(v)
    945∘945^\circ
  12. Ex 1.1 Q1B(vi)
    1120∘1120^\circ
  13. Ex 1.1 Q1B(vii)
    −80∘-80^\circ
  14. Ex 1.1 Q1B(viii)
    −330∘-330^\circ
  15. Ex 1.1 Q1B(ix)
    −500∘-500^\circ
  16. Ex 1.1 Q1B(x)
    −820∘-820^\circ
  17. Convert the following angles in to radian.
    Ex 1.1 Q2(i)
    85∘85^\circ
  18. Ex 1.1 Q2(ii)
    250∘250^\circ
  19. Ex 1.1 Q2(iii)
    −132∘-132^\circ
  20. Ex 1.1 Q2(iv)
    65∘30′65^\circ 30'
  21. Ex 1.1 Q2(v)
    75∘30′75^\circ 30'
  22. Ex 1.1 Q2(vi)
    40∘48′40^\circ 48'
  23. Convert the following angles in degree.
    Ex 1.1 Q3(i)
    7πc12\frac{7\pi^{c}}{12}
  24. Ex 1.1 Q3(ii)
    −5πc3\frac{-5\pi^{c}}{3}
  25. Ex 1.1 Q3(iii)
    5c5^{c}
  26. Ex 1.1 Q3(iv)
    11πc18\frac{11\pi^{c}}{18}
  27. Ex 1.1 Q3(v)
    (−14)c\left(\frac{-1}{4}\right)^{c}
  28. Express the following angles in degree, minute and second.
    Ex 1.1 Q4(i)
    (183.7)∘(183.7)^\circ
  29. Ex 1.1 Q4(ii)
    (245.33)∘(245.33)^\circ
  30. Ex 1.1 Q4(iii)
    (15)c\left(\frac{1}{5}\right)^{c}
  31. Ex 1.1 Q5
    In △\triangle ABC, if m∠A=7πc36m\angle A = \frac{7\pi^{c}}{36}, m∠B=120∘m\angle B = 120^\circ, find m∠Cm\angle C in degree and radian.
  32. Ex 1.1 Q6
    Two angles of a triangle are 5πc9\frac{5\pi^{c}}{9} and 5πc18\frac{5\pi^{c}}{18}. Find the degree and radian measure of third angle.
  33. Ex 1.1 Q7
    In a right angled triangle, the acute angles are in the ratio 4:54:5. Find the angles of the triangle in degree and radian.
  34. Ex 1.1 Q8
    The sum of two angles is 5πc5\pi^{c} and their difference is 60∘60^\circ. Find their measures in degree.
  35. Ex 1.1 Q9
    The measures of the angles of a triangle are in the ratio 3:7:83:7:8. Find their measures in degree and radian.
  36. Ex 1.1 Q10
    The measures of the angles of a triangle are in A.P. and the greatest is 5 times the smallest (least). Find the angles in degree and radian.
  37. Ex 1.1 Q11
    In a cyclic quadrilateral two adjacent angles are 40∘40^\circ and πc3\frac{\pi^{c}}{3}. Find the angles of the quadrilateral in degree.
  38. Ex 1.1 Q12
    One angle of a quadrilateral has measure 2πc5\frac{2\pi^{c}}{5} and the measures of other three angles are in the ratio 2:3:42:3:4. Find their measures in degree and radian.
  39. Find the degree and radian measure of exterior and interior angle of a regular
    Ex 1.1 Q13(i)
    Pentagon
  40. Ex 1.1 Q13(ii)
    Hexagon
  41. Ex 1.1 Q13(iii)
    Heptagon
  42. Ex 1.1 Q13(iv)
    Octagon
  43. Find the angle between hour-hand and minute-hand in a clock at
    Ex 1.1 Q14(i)
    ten past eleven
  44. Ex 1.1 Q14(ii)
    twenty past seven
  45. Ex 1.1 Q14(iii)
    thirty five past one
  46. Ex 1.1 Q14(iv)
    quarter to six
  47. Ex 1.1 Q14(v)
    2:20
  48. Ex 1.1 Q14(vi)
    10:10

1.2 Arc Length and Area of a Sector

6 q

Solved Examples

Worked · 6
  1. 1.2 SolvedEx.1
    The diameter of a circle is 14 cm. Find the length of the arc, subtending an angle of 54∘54^\circ at the centre.
  2. 1.2 SolvedEx.2
    In a circle of radius 12 cms, an arc PQ subtends an angle of 30∘30^\circ at the centre. Find the area between the arc PQ and chord PQ.
  3. 1.2 SolvedEx.3
    The area of a circle is 225π225\pi sq. cm. Find the length of its arc subtending an angle of 120∘120^\circ at the centre. Also find the area of the corresponding sector.
  4. 1.2 SolvedEx.4
    The perimeter of a sector is equal to half of the circumference of a circle. Find the measure of the angle of the sector at the centre in radian.
  5. 1.2 SolvedEx.5
    A pendulum of length 21 cm oscillates through an angle of 36∘36^\circ. Find the length of its path.
  6. 1.2 SolvedEx.6
    ABCDEFGH is a regular octagon inscribed in a circle of radius 9 cm. Find the length of minor arc AB.

Exercise 1.2

10 q
  1. Ex 1.2 Q1
    Find the length of an arc of a circle which subtends an angle of 108∘108^\circ at the centre, if the radius of the circle is 15 cm.
  2. Ex 1.2 Q2
    The radius of a circle is 9 cm. Find the length of an arc of this circle which cuts off a chord of length, equal to length of radius.
  3. Ex 1.2 Q3
    Find the angle in degree subtended at the centre of a circle by an arc whose length is 15 cm, if the radius of the circle is 25 cm.
  4. Ex 1.2 Q4
    A pendulum of length 14 cm oscillates through an angle of 18∘18^\circ. Find the length of its path.
  5. Ex 1.2 Q5
    Two arcs of the same lengths subtend angles of 60∘60^\circ and 75∘75^\circ at the centres of two circles. What is the ratio of radii of two circles ?
  6. Ex 1.2 Q6
    The area of a circle is 25π25\pi sq.cm. Find the length of its arc subtending an angle of 144∘144^\circ at the centre. Also find the area of the corresponding sector.
  7. Ex 1.2 Q7
    OAB is a sector of the circle having centre at O and radius 12 cm. If m∠AOB=45∘m\angle AOB = 45^\circ, find the difference between the area of sector OAB and triangle AOB.
  8. Ex 1.2 Q8
    OPQ is the sector of a circle having centre at O and radius 15 cm. If m∠POQ=30∘m\angle POQ = 30^\circ, find the area enclosed by arc PQ and chord PQ.
  9. Ex 1.2 Q9
    The perimeter of a sector of the circle of area 25π25\pi sq.cm is 20 cm. Find the area of the sector.
  10. Ex 1.2 Q10
    The perimeter of the sector of the circle of area 64π64\pi sq.cm is 56 cm. Find the area of the sector.

Miscellaneous Exercise 1

21 q

(I) Select the correct option

Practice · 10
  1. Misc I Q1
    (22π15)c\left(\dfrac{22\pi}{15}\right)^{c} is equal to
    1. A.
      246∘246^\circ
    2. B.
      264∘264^\circ
    3. C.
      224∘224^\circ
    4. D.
      426∘426^\circ
  2. Misc I Q2
    156∘156^\circ is equal to
    1. A.
      (17π15)c\left(\dfrac{17\pi}{15}\right)^{c}
    2. B.
      (13π15)c\left(\dfrac{13\pi}{15}\right)^{c}
    3. C.
      (11π15)c\left(\dfrac{11\pi}{15}\right)^{c}
    4. D.
      (7π15)c\left(\dfrac{7\pi}{15}\right)^{c}
  3. Misc I Q3
    A horse is tied to a post by a rope. If the horse moves along a circular path, always keeping the rope tight and describes 88 meter when it traces the angle of 72∘72^\circ at the centre, then the length of the rope is
    1. A.
      70 m.
    2. B.
      55 m.
    3. C.
      40 m.
    4. D.
      35 m.
  4. Misc I Q4
    If a 14cm long pendulum oscillates through an angle of 12∘12^\circ, then find the length of its path.
    1. A.
      13π14\dfrac{13\pi}{14}
    2. B.
      14π13\dfrac{14\pi}{13}
    3. C.
      15π14\dfrac{15\pi}{14}
    4. D.
      14π15\dfrac{14\pi}{15}
  5. Misc I Q5
    Angle between hands of a clock when it shows the time 9.45 is
    1. A.
      (7.5)∘(7.5)^\circ
    2. B.
      (12.5)∘(12.5)^\circ
    3. C.
      (17.5)∘(17.5)^\circ
    4. D.
      (22.5)∘(22.5)^\circ
  6. Misc I Q6
    20 meter of wire is available for fancing off a flower-bed in the form of a circular sector of radius 5 meters, then the maximum area (in sq. m.) of the flower-bed is
    1. A.
      15
    2. B.
      20
    3. C.
      25
    4. D.
      30
  7. Misc I Q7
    If the angles of a triangle are in the ratio 1:2:3, then the smallest angle in radian is
    1. A.
      π3\dfrac{\pi}{3}
    2. B.
      π6\dfrac{\pi}{6}
    3. C.
      π2\dfrac{\pi}{2}
    4. D.
      π9\dfrac{\pi}{9}
  8. Misc I Q8
    A semicircle is divided into two sectors whose angles are in the ratio 4:5. Find the ratio of their areas?
    1. A.
      5:1
    2. B.
      4:5
    3. C.
      5:4
    4. D.
      3:4
  9. Misc I Q9
    Find the measure of the angle between hour-hand and the minute hand of a clock at twenty minute past two.
    1. A.
      50∘50^\circ
    2. B.
      60∘60^\circ
    3. C.
      54∘54^\circ
    4. D.
      65∘65^\circ
  10. Misc I Q10
    The central angle of a sector of circle of area 9π9\pi sq.cm is 60∘60^\circ, the perimeter of the sector is
    1. A.
      π\pi
    2. B.
      3+π3+\pi
    3. C.
      6+π6+\pi
    4. D.
      6

(II)

Practice · 11
  1. Misc II Q1
    Find the number of sides of a regular polygon if each of its interior angle is (3π4)c\left(\dfrac{3\pi}{4}\right)^{c}.
  2. Misc II Q2
    Two circles, each of radius 7 cm, intersect each other. The distance between their centres is 727\sqrt{2} cm. Find the area of the portion common to both the circles.
  3. Misc II Q3
    Δ\Delta PQR is an equilateral triangle with side 18 cm. A circle is drawn on the segment QR as diameter. Find the length of the arc of this circle within the triangle.
  4. Misc II Q4
    Find the radius of the circle in which central angle of 60∘60^\circ intercepts an arc of length 37.4 cm.
  5. Misc II Q5
    A wire of length 10 cm is bent so as to form an arc of a circle of radius 4 cm. What is the angle subtended at the centre in degrees ?
  6. Misc II Q6
    If two arcs of the same length in two circles subtend angles 65∘65^\circ and 110∘110^\circ at the centre. Find the ratio of their radii.
  7. Misc II Q7
    The area of a circle is 81π81\pi sq.cm. Find the length of the arc subtending an angle of 300∘300^\circ at the centre and also the area of the corresponding sector.
  8. Misc II Q8
    Show that minute hand of a clock gains 5∘30′5^\circ 30' on the hour hand in one minute.
  9. Misc II Q9
    A train is running on a circular track of radius 1 km at the rate of 36 km per hour. Find the angle to the nearest minute, through which it will turn in 30 seconds.
  10. Misc II Q10
    In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.
  11. Misc II Q11
    The angles of a quadrilateral are in A.P. and the greatest angle is double the least. Find angles of the quadrilateral in radian.