A Doorway is Decorated as Shown in the Figure. There Are Four Semi-circles. Bc, the Diameter of the Larger Semi-circle is of Length 84 Cm. Centres of the Three Equal Semicircles Lie on Bc. Abc is a - Mathematics

Advertisements
Advertisements
Sum

A doorway is decorated as shown in the figure. There are four semi-circles. BC, the diameter of the larger semi-circle is of length 84 cm. Centres of the three equal semicircles lie on BC. ABC is an isosceles triangle with AB = AC. If BO = OC, find the area of the shaded region. (Take `pi = 22/7`)

Advertisements

Solution 1

Here, the radius of larger semicircle = `84/2 = 42 cm`

And, the radius of smaller semi-circle = `84/(3xx2) = 14 cm`

Area of the shaded region  =` (pi(42)^2)/2 + 3xx(pi(14)^2)/2 - 1/2 xx 84 xx 42`

`= 22/7 [21 xx 42 + 3 xx 14 xx 7] - 42 xx 42`

`= 22[3 xx 42 + 42] - 42 xx 42`

`= 42 xx [88 - 42]`

`= 1932 cm^2`

Solution 2

As angle in a semicircle is 90°,
∠ A = 90°
From Δ ABC,
by Pythagoras theorem, we get
AB2 + AC2 = BC2
⇒ x2 + x2 = 842
⇒ 2x2 = 84 x 84
⇒ x2 = 84 x 42

∴ Area of Δ ABC = `1/2` x AB x AC
= `1/2` x 84 cm x 42 cm
= 1764 cm2

Radius of semicircle with BC as diameter = `1/2` x 84 = 42 cm

Diameter of each of three equal semicircles = `1/3` x 84 = 28 cm

⇒ Radius of each of 3 equal semicircles = 14 cm

Area of the shaded region = Area of semicircle with 42 cm as Radius + Area of three equal semicircles of radius 14 cm - area of Δ ABC

= `1/2π xx 422 cm2 + 3 xx 1/2π xx 142 cm2 - 1764 cm2

= `1/2` π ( 422 + 3 xx 142 ) cm2 - 1764 cm2

= `1/2 xx 22/7` x 142 ( 9 + 3) cm2 - 1764 cm2

= 3696 cm2 - 1764 cm= 1932 cm2

  Is there an error in this question or solution?
Chapter 17: Mensuration - Exercise 3

APPEARS IN

ICSE Class 10 Mathematics
Chapter 17 Mensuration
Exercise 3 | Q 10

Video TutorialsVIEW ALL [2]

RELATED QUESTIONS

The radius of a circle is 7 cm. find the circumference of the circle.


In figure, ΔABC is an isosceles triangle with perimeter 44 cm. The base BC is of length 12 cm. Side AB and side AC are congruent. A circle touches the three sides as shown in the figure below. Find the length of the tangent segment from A to the circle.


A bicycle whell makes 5000 revolutions in moving 11 km. Find the diameter of the wheel


Saima wants to put a lace on the edge of a circular table cover of diameter 1.5 m. Find the length of the lace required and also find its cost if one meter of the lace costs ₹ 15. ( Take π = 3.14)


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 = 22/7`)


ABC is an isosceles right-angled triangle with ∠ABC = 90°. A semi-circle is drawn with AC as the diameter. If AB = BC = 7 cm, find the area of the shaded region. [Take π = 22/7]


Find in terms of x the length of the arc that subtends an angle of 30°, at the centre of circle of radius 4 cm.


The minute hand of clock is10cm long. Find the area of the face of the clock described by the minute hand between 8am and 8:25 am


A square water tank has its side equal to 40m, there are 4 semicircular flower beds grassy plots all around it. Find the cost of turfing the plot at Rs 1.25/sq.m


The area of rhombus is  `480cm^2` , and one of its diagonal measures 48 cm. Find

(i) the length of the other diagonal,

(ii) the length of each of the sides

(iii) its perimeter 

 


In the given figure, the radius of the circle is 7 cm and m ( arc MBN) = 60°,
find  A(O - MCN). 

In the given figure, O is the centre of the sector. \[\angle\]ROQ =  \[\angle\]MON = 60° . OR = 7 cm, and   OM = 21 cm. Find the lengths of  arc RXQ and arc MYN. (\[\pi = \frac{22}{7}\]) 


In the given figure, if A(P-ABC) = 154 cm2  radius of the circle is 14 cm, find

(1) `∠APC` 

(2) l ( arc ABC) .


The radius of a sector of a circle is 7 cm. If the measure of the arc of the sector is - 30°  find the area of the sector in case.

The radius of a sector of a circle is 7 cm. If the measure of the arc of the sector is 210°,  find the area of the sector in case.


The area of a minor sector of a circle is 3.85 cmand the measure of its central angle is 36°. Find the radius of the circle.


Choose the correct alternative answer for the following question.
The ratio of circumference and area of a circle is 2:7. Find its circumference.

Choose the correct alternative answer for the following question.

The curved surface area of a cylinder is 440 cm 2 and its radius is 5 cm. Find its height.


Choose the correct alternative answer for the following question.

A cone was melted and cast into a cylinder of the same radius as that of the base of the cone. If the height of the cylinder is 5 cm, find the height of the cone.

 

Choose the correct alternative answer for the following question.

 Find the volume of a cube of side 0.01 cm.

Choose the correct alternative answer for the following question.

Find the side of a cube of volume 1 m3.

Diagonal of a square is 20 cm. Find the length and perimeter of the square.


A path of 8 m width runs around the outsider of a circular park whose radius is 17 m. Find the area of the path.


The wheels of the locomotive of a train are 2.1 m in radius. They make 75 revolutions in one minute.


The perimeter of a circular field is 242 m. The area of the field is


The areas of two circles are in the ratio 9 : 4. The ratio of their circumferences is


Radii of the circles are given below, find their areas.
(1) 28 cm
(2) 10.5 cm
(3) 17.5 cm


Areas of some circles are given below find their diameters.
(1) 176 sq cm
(2) 394.24 sq cm
(3) 12474 sq cm


Diameter of the circular garden is 42 m. There is a 3.5 m wide road around the garden. Find the area of the road.


Find the area of the circle if its circumfence is 88 cm.


Find the curved surface area , the total surface area and the volume of a cone if its : 

Height = 12 cm , radius = 5 cm


Find the area and perimeter of the circle with the following :

Radius= 2.8 cm 


Find the area and perimeter of the circle with the following :

 Diameter = 77 cm 


Find the curved surface area , the total surface area and the volume of a cone if its : 

Height = 16 cm , diameter = 24 cm


Find the curved surface area , the total surface area and the volume of a cone if its : 

Height = 8 cm , diameter = 12 cm


Find the circumference of the circle whose area is 25 times the area of the circle with radius 7 cm. 


The circumference of a circle is equal to the perimeter of a square. The area of the square is 484 sq. m. Find the area of the circle. 


Find the area and perimeter of the following semi-circle :

Diameter= 7 cm


A conical tent with a capacity of 600 m3 stands on a circular base of area 160 m2 Find in m2 the area of the canvas. 


Find the area and perimeter of the following semi-circle :

Diameter= 5.6 cm 


A 7 m road surrounds a circular garden whose area is 5544 m2. Find the area of the road and the cost of tarring it at the rate of Rs.150 per sq m.  


A hollow metallic cylindrical tube has an internal radius of 3.5 cm and height 21 cm. The thickness of the metal tube is 0.5 cm. The tube is melted and cast into a right circular cone of height 7 cm. Find the radius of the cone, correct to one decimal place. 


Two circles touch each other externally. The sum of their areas is 5811 cm2 and the distance between their centres is 10 cm. Find the radii of the two circles.


The sum of the radii of two circles is 10.5 cm and the difference of their circumferences is 13. 2 cm. Find the radii of the two circles. 


Sand from a cylindrical bucket 32 cm in height and 18 cm in radius is poured onto the ground making a conical heap 24 cm high. Find the radius of the conical heap. 


Find the volume and the surface area of the spheres in the following :

Radius= 2.1 cm 


Find the diameter of the sphere for the following : 

Surface Area = `576pi`cm2


The diameter of a circle is 28 cm.
Find its :
(i) Circumference
(ii) Area.


An express train is running between two stations with a uniform speed. If the diameter of each wheel of the train is 42 cm and each wheel makes 1200 revolutions per minute, find the speed of the train.


A wheel has a diameter of 84 cm. Find how many completer revolutions must it make to cover 3.168 km.


The circumference of a given circular park is 55 m. It is surrounded by a path of uniform width of 3.5 m. Find the area of the path.


Draw the circumcircle of Δ PMT in which PM = 5.4, P = 60°, M = 70°.


The radius of a circle is 21 cm. Find the circumference (Take π = `3 1/7`).


The circumference of a circle is 440 cm. Find its radius and diameter. (Take π = `22/7`)


The radius of two circles are 20 cm and 13 cm. Find the difference between their circumferences. (Take π =`22/7`)


The perimeter of a square and the circumference of a circle are equal. If the length of each side of the square is 22 cm, find:
(i) perimeter of the square.
(ii) circumference of the circle.
(iii) radius of the circle.


Find the radius of the circle whose circumference is equal to the sum of the circumferences of the circles having radius 15 cm and 8 cm.


The circumference of a circle is eigth time the circumference of the circle with a radius of 12 cm. Find its diameter.


The circumferences of two circles are in the ratio 5: 7, find the ratio between their radius.


Each side of a square is 44 cm. Find its perimeter. If this perimeter is equal to the circumference of a circle, find the radius of the circle.


The circumference of a circle exceeds its diameter by 18 cm. find the radius of the circle.


The inner circumference of a circular track is 264 m and the width of the track is 7 m. Find:
(i) the radius of the inner track.
(ii) the radius of the outer circumference.
(iii) the length of the outer circumference.
(iv) the cost of fencing the outer circumference at the rate of ₹50 per m.


In the adjoining figure, the radius is 3.5 cm. Find:
(i) The area of the quarter of the circle correct to one decimal place.
(ii) The perimeter of the quarter of the circle correct to one decimal place. ( Take π = `22/7`)


Complete the table below.

Radius (r)  Diameter (d) Circumference (c)
7 cm ______ _______

Complete the table below.

Radius (r)  Diameter (d) Circumference (c)
_______ 28 cm _________

Complete the table below.

Radius (r)  Diameter (d) Circumference (c)
_______ _________ 616 cm

Complete the table below.

Radius (r)  Diameter (d) Circumference (c)
_______ _________ 72.6 cm

If the circumference of a circle is 176 cm, find its radius.


The wheel of a bullock cart has a diameter of 1.4 m. How many rotations will the wheel complete as the cart travels 1.1 km?


The diameter of a circle is 5.6 cm. Find its circumference.


The diameter of the wheel of a cart is 1.05 m. How much distance will the cart cover in 1000 rotations of the wheel?


Find the area and perimeter of the circles with following: Radius = 2.8cm


Find the area and perimeter of the circles with following: Diameter = 77cm


Find the circumference of a circle whose area is 81πcm2.


The speed of a car is 66km per hour. If each wheel of the car is 140cm in diameter, find the number if revolution made by each wheel per minute.


A wire when bent in the form of a square encloses an area of 484cm2. If the same wire is bent into the form of a circle, find the area of the circle.


A 4.2m wide road surrounds a circular plot whose circumference is 176m. Find the cost of paving the road at Rs.75per m2.


Sudhanshu divides a circular disc of radius 7 cm in two equal parts. What is the perimeter of each semicircular shape disc? (Use π = `22/7`)


The radius of a circular plot is 7.7 metres. How much will it cost to fence the plot with 3 rounds of wire at the rate of 50 rupees per metre?


The radius of the wheel of a bus is 0.7 m. How many rotations will a wheel complete while traveling a distance of 22 km?


Diameters of different circles are given below. Find their circumference (Take π = `22/7`)

d = 28 mm


Find the circumference of the circles whose radii are given below.

49 cm


Find the circumference of the circles whose radii are given below.

91 mm


A wire of length 1320 cm is made into circular frames of radius 7 cm each. How many frames can be made?


A Rose garden is in the form of circle of radius 63 m. The gardener wants to fence it at the rate of ₹ 150 per metre. Find the cost of fencing?


In the formula, C = 2πr, ‘r’ refers to


Circumference of a circle is always ______.


A wheel of a car covers a distance of 3520 cm in 20 rotations. Find the radius of the wheel?


The cost of fencing a circular race course at the rate of ₹ 8 per metre is ₹ 2112. Find the diameter of the race course


The cost of decorating the circumference of a circular lawn of a house at the rate of ₹ 55 per metre is ₹ 16940. What is the radius of the lawn?


A circular pond is 17.5 m is of diameter. It is surrounded by a 2 m wide path. Find the cost of constructing the path at the rate of Rs 25 per m2.


The area of a semicircle of radius 4r is ______.


The distance around a circle is its ______.


Ratio of circumference of a circle to its radius is always 2π:I.


A circle with radius 16 cm is cut into four equal parts and rearranged to form another shape as shown in the below figure:


Does the perimeter change? If it does change, by how much does it increase or decrease?


ABCD is a given rectangle with length as 80 cm and breadth as 60 cm. P, Q, R, S are the midpoints of sides AB, BC, CD, DA respectively. A circular rangoli of radius 10 cm is drawn at the centre as shown in the given figure. Find the area of shaded portion.


A dinner plate is in the form of circle. A circular region encloses a beautiful design as shown in the given figure. The inner circumference is 352 mm and outer is 396 mm. Find the width of circular design.


The moon is about 384000 km from earth and its path around the earth is nearly circular. Find the length of path described by moon in one complete revolution. (Take π = 3.14)


Share
Notifications



      Forgot password?
Use app×