Mathematics · Class 11 Science
Ch 1Angle and its Measurement — Class 11 Mathematics, concept-first.
In elementary geometry an angle is described just by its two arms, with no sense of "which way" it opens -- an angle of drawn turning anticlockwise looks the same size as one drawn turning clockwise, even though the two constructions are genuinely different.
Key concepts
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Directed Angles and Standard Position
A directed angle is formed by rotating an initial ray about its endpoint until it reaches a terminal ray ; the rotation's sense (anticlockwise = positive, clockwise = negative) is part of the angle's identity, so in gene…
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Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Directed Angles
In elementary geometry an angle is described just by its two arms, with no sense of "which way" it opens -- an angle of drawn turning anticlockwise looks the same size as one drawn turning clockwise,…
+−Exercise 1.1i15 questions
- Q1Determine which of the following pairs of angles are co-terminal. i) 210°, −150° ii) 360°, −30° iii) −180°, 540° iv) −405°, 675° v) 860°, 58…Free
- Q2Draw the angles of the following measures and determine their quadrants. i) –140° ii) 250° iii) 420° iv) 750° v) 945° vi) 1120° vii) –80° vi…Free
- Q3Convert the following angles into radian. i) 85° ii) 250° iii) −132° iv) 65°30′ v) 75°30′ vi) 40°48′Free
- Q4Convert the following angles into degree. i) $\dfrac{7\pi}{12}^{c}$ ii) $\dfrac{-5\pi}{3}^{c}$ iii) $5^{c}$ iv) $\dfrac{11\pi}{18}^{c}$ v) $…Preview
- Q5Express the following angles in degree, minute and second. i) $(183.7)°$ ii) $(245.33)°$ iii) $\dfrac{1}{5}^{c}$Preview
- Q6In $\triangle ABC$, if $m\angle A = \dfrac{7\pi}{36}^{c}$, $m\angle B = 120°$, find $m\angle C$ in degree and radian.Preview
- Q7Two angles of a triangle are $\dfrac{5\pi}{9}^{c}$ and $\dfrac{5\pi}{18}^{c}$. Find the degree and radian measure of the third angle.Preview
- Q8In a right angled triangle, the acute angles are in the ratio 4:5. Find the angles of the triangle in degree and radian.Preview
- Q9The sum of two angles is $5\pi^{c}$ and their difference is 60°. Find their measures in degree.Preview
- Q10The measures of the angles of a triangle are in the ratio 3:7:8. Find their measures in degree and radian.Preview
- Q11The 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.Preview
- Q12In a cyclic quadrilateral two adjacent angles are 40° and $\dfrac{\pi}{3}^{c}$. Find the angles of the quadrilateral in degree.Preview
- Q13One angle of a quadrilateral has measure $\dfrac{2\pi}{5}^{c}$ and the measures of other three angles are in the ratio 2:3:4. Find their mea…Preview
- Q14Find the degree and radian measure of exterior and interior angle of a regular i) Pentagon ii) Hexagon iii) Heptagon iv) OctagonPreview
- Q15Find the angle between hour-hand and minute-hand in a clock at i) ten past eleven ii) twenty past seven iii) thirty five past one iv) quarte…Preview
Measures of angles
Once a directed angle has been defined by the amount and sense of rotation carrying the initial arm onto the terminal arm, that amount of rotation still needs a numerical measure so it can be stated,…
Sexagesimal System (Degree Measure)
In the sexagesimal ("base-60") system, the unit of angle measurement is the degree. One complete rotation is divided into equal parts, and the measure of each part is called one degree, written :
Circular System (Radian Measure)
In the circular system, the unit of angle measurement is the radian. Let a circle have centre and radius , and let , be two points on the circle such that the length of arc is exactly equal to .
Arc Length and Area of a Sector
A sector of a circle is the region enclosed between two radii and the arc they cut off. Both the length of that arc and the area of that sector are directly proportional to the size of the central ang…
+−Exercise 1.2i10 questions
- Q16Find the length of an arc of a circle which subtends an angle of $108°$ at the centre, if the radius of the circle is 15 cm.Free
- Q17The 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.Free
- Q18Find 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.Free
- Q19A pendulum of length 14 cm oscillates through an angle of $18°$. Find the length of its path.Preview
- Q20Two arcs of the same lengths subtend angles of $60°$ and $75°$ at the centres of two circles. What is the ratio of radii of two circles?Preview
- Q21The area of a circle is $25\pi$ sq.cm. Find the length of its arc subtending an angle of $144°$ at the centre. Also find the area of the cor…Preview
- Q22OAB is a sector of the circle having centre at O and radius 12 cm. If $m\angle AOB = 45°$, find the difference between the area of sector OA…Preview
- Q23OPQ is the sector of a circle having centre at O and radius 15 cm. If $m\angle POQ = 30°$, find the area enclosed by arc PQ and chord PQ.Preview
- Q24The perimeter of a sector of the circle of area $25\pi$ sq.cm is 20 cm. Find the area of the sector.Preview
- Q25The perimeter of the sector of the circle of area $64\pi$ sq.cm is 56 cm. Find the area of the sector.Preview
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21 Q+−Show 10 questionsHide questions10 questions
- Q26$\dfrac{22\pi}{15}^{c}$ is equal to A) $246°$ B) $264°$ C) $224°$ D) $426°$Free
- Q27$156°$ is equal to A) $\dfrac{17\pi}{15}^{c}$ B) $\dfrac{13\pi}{15}^{c}$ C) $\dfrac{11\pi}{15}^{c}$ D) $\dfrac{7\pi}{15}^{c}$Free
- Q28A 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…Free
- Q29If a 14 cm long pendulum oscillates through an angle of $12°$, then find the length of its path. A) $\dfrac{13\pi}{14}$ B) $\dfrac{14\pi}{13…Preview
- Q30Angle between hands of a clock when it shows the time 9.45 is A) $(7.5)°$ B) $(12.5)°$ C) $(17.5)°$ D) $(22.5)°$Preview
- Q3120 meter of wire is available for fencing off a flower-bed in the form of a circular sector of radius 5 meters, then the maximum area (in sq…Preview
- Q32If the angles of a triangle are in the ratio 1:2:3, then the smallest angle in radian is A) $\dfrac{\pi}{3}$ B) $\dfrac{\pi}{6}$ C) $\dfrac{…Preview
- Q33A semicircle is divided into two sectors whose angles are in the ratio 4:5. Find the ratio of their areas? A) 5:1 B) 4:5 C) 5:4 D) 3:4Preview
- Q34Find the measure of the angle between hour-hand and the minute hand of a clock at twenty minute past two. A) $50°$ B) $60°$ C) $54°$ D) $65°…Preview
- Q35The central angle of a sector of circle of area $9\pi$ sq.cm is $60°$, the perimeter of the sector is A) $\pi$ B) $3+\pi$ C) $6+\pi$ D) $6$Preview
+−Show 11 questionsHide questions11 questions
- Q36Find the number of sides of a regular polygon if each of its interior angle is $\dfrac{3\pi}{4}^{c}$.Free
- Q37Two circles, each of radius 7 cm, intersect each other. The distance between their centres is $7\sqrt{2}$ cm. Find the area of the portion c…Free
- Q38$\triangle PQR$ 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 t…Free
- Q39Find the radius of the circle in which central angle of $60°$ intercepts an arc of length 37.4 cm.Preview
- Q40A 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?Preview
- Q41If two arcs of the same length in two circles subtend angles $65°$ and $110°$ at the centre. Find the ratio of their radii.Preview
- Q42The area of a circle is $81\pi$ sq.cm. Find the length of the arc subtending an angle of $300°$ at the centre and also the area of the corre…Preview
- Q43Show that minute hand of a clock gains $5°30'$ on the hour hand in one minute.Preview
- Q44A 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…Preview
- Q45In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.Preview
- Q46The angles of a quadrilateral are in A.P. and the greatest angle is double the least. Find angles of the quadrilateral in radian.Preview