Areas Related to Circles Chapter Wise Important Questions Class 10 Mathematics

Areas Related to Circles Chapter Wise Important Questions Class 10 Mathematics

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2016

Short Answer Type Questions II [3 Marks]

Question 1.
In figure, ABCD is a square of side 14 cm. Semi-circles are drawn with each side of square as diameter. Find the area of the shaded region.

Solution:
Area of the square ABCD = 14 x 14 = 196 cm²
Area of semicircle AOB=1/2 x πr²
=1/2×22/7x7x7
Similarly, area of semicircle DOC = 77 cm²
Hence, the area of shaded region (Part W and Part Y) = Area of square -Area of two semicircles AOB and COD
= 196 – 154 = 42 cm²
Therefore, area of four shaded parts (i.e. X, Y, W, Z) = (2 x 42) cm² = 84 cm²

Question 2.
In figure, are shown two arcs PAQ and PBQ. Arc PAQ is a part of circle with centre O and radius OP while arc PBQ is a semicircle drawn on PQ as diameter with centre M. If OP = PQ = 10 cm, show that area of shaded region is
25(√3- π /6)cm²

Solution:
OP = OQ = 10 cm
PQ = 10 cm
So, ΔOPQ is an equilateral triangle
∠POQ = 60°
Area of segment PAQM = Area of sector OPAQ – Area of ΔOPQ
=60/360 x π x 10 x 10-√3/4 x 10 x 10
=(100π/6-100√3/4)cm²
Area of semicircle =1/2 x π x 5 x 5 = 25/2π cm²
Area of the shaded region =25/2π-(100π/6-100√3/4)=25π/2-50π/3+25√3
=75π-100π/6+25√3=25√3-25π/6
=25(√3-π/6)cm²

Question 3.
In figure, O is the centre of a circle such that diameter AB = 13 cm and AC = 12 cm. BC is joined. Find the area of the shaded region.

Solution:
Here, BC² = AB²-AC²
= 169-144 = 25 BC = 5
Area of shaded region = Area of semicircle – Area of right triangle ABC
=1/2 x πr²-1/2AC x BC
=1/2 x 3.14(13/2)²-1/2 x 12 x 5
=66.33 – 30 = 36.33 cm²

Question 4.
In figure, find the area of the shaded region, enclosed between two concentric circles of radii 7 cm and 14 cm where∠AOC = 40°.

Solution:
Area of shaded region=360°-θ/360° x π(R²-r²)
=320°/360° x π[(14)²-(7)²]
=8/9 x 22/7(196-49)=8/9 x 22/7 x 147
=1232/3=410.67 cm²

Question 5.
Find the area of shaded region in figure, where a circle of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm

Solution:
Area of  ΔOAB=√3/4(side)²=√3/4 x (12)²
=36√3=36 x 1.73
=62.28 cm²
area of circle with center O =πr²=3.14 x (6)²
=3.14 x 36 =113.04 cm²
area of sector(OLQP)=πr² x θ/360°=3.14 x 6² x 60°/360°
=3.14 x 36 x 1/6 =18.84 cm²
area of shaded region=area of ΔOAB+area of circle-2 area of sector OLQP
=(62.28+113.04-2 x 18.84)cm²
=137.64cm²

Question 6.
In figure, is a chord AB of a circle, with centre O and radius 10 cm, that subtends a right angle at the centre of the circle. Find the area of the minor segment AQBP. Hence, find the area of major segment ALBQA

Solution:
Area of minor segment APBQ=θ/360° x πr²-r²sin45°cos45°
=3.14 x 100/4-100 x 1/√2 x 1/√2
=(78.5-50)cm²=28.5 cm²
Area of major segment ALBQA =πr²-area of minor segment
=3.14 x (10)²-28.5
=(314-28.5)cm²=285.5 cm²

Long Answer Type Questions [4 Marks]

Question 7.
An elastic belt is placed around the rim of a pulley of radius 5 cm. From one point C on the belt, the elastic belt is pulled directly away from the centre O of the pulley until it is at P, 10 cm from the point O. Find the length of the belt that is still in contact with the pulley.Also find the shaded area, (use π = 3.14 and √3 = 1.73)

Solution:

Question 8.
In figure, is shown a sector OAP of a circle with centre O, containing Z0. AB is perpendicular to the radius OA and meets OP produced at B. Prove that the perimeter of shaded region is r[tanθ+secθ+πθ/180-1]

Solution:

Question 9.
Find the area of the shaded region in figure, where APD, AQB, BRC and CSD are semi-circles of diameter
14 cm, 3.5 cm, 7 cm and 3.5 cm respectively

Solution:
Area of shaded region
= Area of semicircle APD + Area of semicircle BRC – 2 x Area of semicircle AQB

2015

Short Answer Type Questions II [3 Marks

Question 10.
In figure, APB and AQO are semicircle, and AO = OB. If the perimeter of the figure is 40 cm, find the area of the shaded region.

Solution:
Let r be the radius of the semicircle APB, i.e. OB = OA = r, then r/2 is the radius of the semicircle AQO.

Question 11.
In figure, find the area of the shaded region

Solution:

Question 12.
Find the area of the minor segment of a circle of radius 14 cm, when its central angle is 60°. Also find the area of the corresponding major segment
Solution:

Question 13.
All the vertices of a rhombus lie on a circle.Find the area of rhombus,if the area of circle is 1256 cm²
Solution:

Question 14.
The long and short hand of a clock are 6cm and 4 cm long respectively,Find the sum of the distance travelled by their tips in 24hrs.
Solution:

Question 15.



Solution:

Long Answer Type Questions [4 Marks]

Question 16.
In Figure, PQRS is a square lawn with side PQ = 42 metres. Two circular flower beds are there on the sides PS and QR with centre at O, the intersection of its diagonals. Find the total area of the two flower beds (shaded parts).

Solution:

2014

Short Answer Type Questions I [2 Marks]

Question 17.
In figure, a square OABC is inscribed in a quadrant OPBQ of a circle. If OA = 20 cm, find the area of the shaded region

Solution:

Question 18.
In figure, OABC is a quadrant of a circle of radius 7 cm. If OD = 4 cm, find the area of the shaded region

Solution:

 

Short Answer Type Questions II [3 Marks]

Question 19.
In figure, a circle is inscribed in an equilateral triangle ABC of side 12 cm. Find the radius of inscribed circle and the area of the shaded region

Solution:

Question 20.
In figure, PSR, RTQ and PAQ are three semicircles of diameters 10 cm, 3 cm and 7 cm respectively. Find the perimeter of the shaded region

Solution:

Question 21.
In figure, two concentric circles with centre O, have radii 21 cm and 42 cm. If ∠AOB = 60°, find the area of the shaded region

Solution:

Question 22.
In figure, ABCD is a trapezium of area 24.5 sq. cm. In it, AD ¦ ¦ BC, ∠DAB = 90°, AD = 10 cm and BC = 4 cm. If ABE is a quadrant of a circle, find the area of the shaded region

Solution:

Question 23.
In figure, ABDC is a quadrant of a circle of radius 28 cm and a semicircle BEC is drawn with BC as diameter. Find B the area of the shaded region.

Solution:

2013

Short Answer Type Questions I [2 Marks]

Question 24.
Two circular pieces of equal radii and maximum area,touching each other are cut out from a rectangular cardboard of dimensions 14 cm x 7 cm. Find the area of the remaining cardboard
Solution:

Question 25.
The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.
Solution:

Question 26.
In the given figure,the area of the shaded region between two concentric circles is 286 cm2.If the difference of the radii of the two circles is 7 cm, find the sum of their radii.

Solution:

Short Answer Type Questions II [3 Marks]

Question 27.
In the given figure AB and CD are two diameters of a circle with centre O,Which are perpendicular to each other.OB is the diameter of small circle.If OA=7cm,find area of shaded region

Solution:

Question 28.
In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find

1. the length of the arc
2. area of the sector formed by the arc.

Solution:

Question 29.
In figure, a square OABC is inscribed in a quadrant OPBQ of a circle. If OA = 21 cm, find the area of the shaded region

Solution:

Question 30.
A chord of length 10 cm divides a circle of radius 5√2 cm in two segments. Find the area of the minor segment
Solution:

Question 31.
In the given figure, from each corner of a square of side 4 cm, a quadrant of a circle of radius 1 cm is cut and also a circle of diameter 2 cm is cut. Find the area of the shaded region.

Solution:

2012

Short Answer Type Questions I [2 Marks]

Question 32.
In figure, OABC is a square of side 7cm. If OAPC is a quadrant of a circle with centre O, then find the area of the shaded region

Solution:

Question 33.
In figure, ABCD is a square of side 4 cm. A quadrant of a circle of radius 1 cm is drawn at each vertex of the square and a circle of diameter 2 cm is also drawn. Find the area of the shaded region

Solution:
Refer to Ans 31.

Question 34.
From a rectangular sheet of paper ABCD with AB = 40 cm and AD = 28 cm, semicircular portion with BC as diameter is cut off. Find the area of the remaining paper.
Solution:

Question 35.
In the given figure, the shape of the top of a table is that a sector of a circle with centre O and ∠AOB = 90°. If AO = OB = 42 cm,then find the perimeter of the top of the table

Solution:

Short Answer Type Questions II [3 Marks]

Question 36.
In figure, PQ and AB are respectively the arcs of two concentric circles of radii 7 cm and 3.5 cm and centre O. If ∠POQ = 30°, then find the area of the shaded region.

Solution:

Question 37.
In figure, find the area of the shaded region, if ABCD is a square of side 14 cm and APD and BPC are semicircles.
Solution:

Question 38.
In the given figure, O is the centre of the circle with AC = 24 cm, AB = 7 cm and ∠BOD = 90°. Find the area of the shaded region.
Solution:

Question 39.
Find the area of the shaded region in Figure, if ABCD is a square of side 28 cm and APD and BPC are semicircles.

Solution:

Question 40.
In figure, ABCD is a square of side 7 cm. DPBA and DQBC are quadrants of circles, each of radius 7 cm. Find the area of the shaded region.

Solution:
side of square=7cm
radius of circle=7cm

Question 41.
The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 10 minutes
Solution:

2011

Short Answer Type Questions I [2 Marks]

Question 42.
In figure APB and CQD are semicircles of diameter 7 cm each, while ARC and BSD are semicircles of diameter 14 cm each. Find the perimeter of the shaded region

Solution:

Question 43.
Find the area of a quadrant of a circle, where the circumference of circle is 44cm
Solution:

Question 44.
Find the perimeter of the shaded region in figure, if ABCD is a square of side 14 cm and APB and CPD are semicircles

Solution:

Question 45.
In given figure, a semicircle is drawn with O as centre and AB as diameter. Semicircles are drawn with AO and OB as diameters.If AB = 28 m, find the perimeter of the shaded region

Solution:

Question 46.
In given figure, ABC is a triangle right-angled atB, with AB = 14 cm and BC = 24 cm. With thevertices A, B and C as centres, arcs are drawn each of radius 7 cm. Find the area of the shaded region.

Solution:

Question 47.
In the given figure, OABC is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the shaded region

Solution:

Short Answer Type Questions II [3 Marks]

Question 48.
Find the area of the major segment APB in figure of a circle of radius 35 cm and ∠AOB = 90°.

Solution:

Question 49.
A chord of a circle of radius 14 cm subtends an angle of 120° at the centre. Find the area of the corresponding minor segment of the circle.
Solution:

Question 50.
A chord of a circle of radius 21 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor segment of the circle
Solution:

Question 51.
Area of a sector of a circle of radius 14 cm is 154 cm². Find the length of the corresponding arc of the sector
Solution:

Question 52.
A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding minor segment and hence find the area of the major segment
Solution:

Long Answer Type Questions [4 Marks]

Question 53.
In figure, arcs are drawn by taking vertices A, B and C of an equilateral triangle ABC of side 14 cm as centres to intersect the sides BC, CA and AB at their respective mid-point D, E and E Find the area of the shaded region

Solution:

Question 54.
The length and breadth of a rectangular piece of paper are 28 cm and 14 cm respectively. A semi-circular portion is cut off from the breadth’s side and a semicircular portion is added on length’s side, as shown in figure. Find the area of shaded region

Solution:

Question 55.
Find the area of shaded region in figure,where arcs drawn with centres A,B,C and D intersects at mid point P,Q,R and S of sides AB,BC,CD and DA of a square ABCD,where the length of each side of square is 14 cm

Solution:

Question 56.
In the given figure, three circles each of radius 3.5 cm are drawn in such a way that each of them touches the other two. Find the area of shaded region enclosed between these three circles

Solution:

Question 57.
In given figure, an equilateral triangle has been inscribed in a circle of radius 6 cm. Find the area of the shaded region.

Solution:

Question 58.
Find the area of the shaded region in given figure, where ABCD is a square of side 28 cm

Solution:

Question 59.
From a thin metallic piece, in the shape of a trapezium ABCD in which AB ¦¦ CD and ∠BCD = 90°, a quarter circle BFEC is removed (See figure). Given AB = BC = 3.5 cm and DE = 2 cm, calculate the area of the remaining (shaded) part of the metal sheet.

Solution:

2010

Short Answer Type Questions II [3 Marks]

Question 60.
In figure, the boundary of shaded region consists of four semicircular arcs, two smallest being equal. If diameter of the largest is 14 cm and that of the smallest is 3.5 cm, calculate the area of the shaded region.

Solution:

Question 61.
Find the area of the shaded region in figure, if AC = 24 cm, BC = 10 cm and O is the centre of the circle

Solution:

Question 62.
In figure, AB and CD are two perpendicular diameters of a circle with centre O. If OA = 7 cm, find the area of the shaded region

Solution:

Question 63.
Find the area of the shaded region in figure, where a circular arc of radius 7 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm, as centre

Solution: