Question 1: Find the circumference and area of a circle of radius .
Answer:
Radius
Circumference of circle
Area of circle
Question 2: Find the circumference of a circle whose area is .
Answer:
Area
Therefore
Therefore circumference of the circle
Question 3: Find the area of a circle whose circumference is .
Answer:
Circumference
Therefore
Therefore Area
Question 4: The circumference of a circle exceeds the diameter by . Find the circumference of the circle.
Answer:
Given:
Therefore Circumference
Question 5: A horse is tied to a pole with long string. Find the area where the horse can graze.
Answer:
Area that the horse can graze
Question 6: A steel wire when bent in the form of a square encloses an area of . If the same wire is bent in the form of a circle, find the area of the circle.
Answer:
Area of square
Let the side of the square
Therefore Perimeter
Let be the radius of the circle
Therefore
Therefore Area of circle
Question 7: The diameters of the front and rear wheels of a tractor are and
respectively. Find the number of revolutions that rear wheel will make to cover the distance which the front wheel covers in
revolutions.
Answer:
Diameter of front wheel
Diameter of rear wheel
Distance covered by front wheel
Let no of revolution that the rear wheel
Therefore
Question 8: A copper wire when bent in the form of a square encloses an area of . lf the same wire is bent into the form of a circle, find the area of the circle.
Answer:
Area of square
Let the side of the square
Therefore Perimeter
Let be the radius of the circle
Therefore
Therefore Area of circle
Question 9: The circumference of two circles are in the ratio . Find the ratio of their areas.
Answer:
Let the radius of first circle
Let the radius of second circle
Therefore
Therefore Ratios of their areas
Question 10: The side of a square is . Find the area of circumscribed and inscribed circles.
Answer:
When square is inscribed in the circle
Therefore diameter
Therefore radius of circle
Therefore are of circle
When the circle is inscribed in the square
Diameter
Therefore radius
Therefore Area of circle
Question 11: The sum of the radii of two circles is and the difference of their circumferences is
. Find the diameters of the circles.
Answer:
Let the two radii be and
Therefore … … … … … i)
… … … … … ii)
From i) and ii)
Therefore
Therefore
Diameter of circles are and
Question 12: Find the area of the circle in which a square of area is inscribed.
Take
Answer:
Area of square
Side of square
Diameter of circle
Radius of the circle
Hence Area of circle
Question 13: A field is in the form of a circle. A fence is to be erected around the field. The cost of fencing would be at the rate of
per meter. Then, the field is to be thoroughly ploughed at the cost of
. What is the amount required to plough the field?
Take
Answer:
Cost of fencing
Circumference of the field
Therefore
Area of the field
Therefore cost of ploughing
Question 14: If square is inscribed in a circle, find the ratio of the areas of the circle and the square.
Answer:
Let the side of the square
Therefore diameter of circle
Therefore radius of the circle
Therefore ratios of their areas
Hence the ratios of their area is
Question 15: A park is in the form of a rectangle . At the center of the park there is a circular lawn. The area of park excluding lawn is
. Find the radius of the circular lawn.
Take
Answer:
Let the radius of the circle
Area of the park
Are of circle
Therefore
Therefore
Question 16: The radii of two circles are and
respectively. Find the radius of the circle having its area equal to the sum of the areas of the two circles.
Answer:
Let the radius of the circle
Therefore
Question 17: The radii of two circles are and
respectively. Find the radius and area of the circle which has its circumference equal to the sum of the circumferences of the two circles.
Answer:
Let the radius of the circle
Therefore
Therefore Area
Circumference
Question 18: A car travels 1-kilometer distance in which each wheel makes complete revolutions. Find the radius of its wheels.
Answer:
Let the radius of the wheel
Therefore
Question 19: The area enclosed between the concentric circles is . If the radius of the outer circle is
, find the radius of the inner circle.
Answer:
Let the radius of the inner circle
Therefore
Therefore
Question 20: The wheel of a car is making revolutions per second. If the diameter of the wheel is
, find its speed in
. Give your answer, correct to nearest km.
Answer:
No. of revolutions per second
Radius of the wheel
Therefore circumference of the wheel
Hence the distance covered in 1 second
Hence distance covered in one hour
Question 21: A sheet is long and
wide. Circular pieces of
in diameter are cut from it to prepare discs. Calculate the number of discs that can be prepared.
Answer:
Number of disks by length
No of disks by breadth
Therefore total number of disks that can be cut
Question 22: A copper wire when bent in the form of an equilateral triangle has area . If the same wire is bent into the form of a circle, find the area enclosed.
Answer:
Let the side of the equilateral triangle
Therefore
Therefore
Therefore perimeter
Let the radius of the circle
Therefore
Area of circle
Question 23: A plot is in the form of a rectangle having semi-circle-on
as shown in the adjoining figure.
and
, find the area of the plot.
Answer:
Area of rectangle
Diameter of semi circle
Therefore Area of semi circle
Hence area of the park
Question 24: A play ground has the shape of a rectangle, with two semi-circles on its smaller sides as diameters, added to its outside. If the sides of the rectangle are and
, find, the area of the playground.
Take
Answer:
Area of rectangle
Radius of semicircle
Therefore Area of 2 semi circles
Therefore total area
Question 25: The outer circumference of a circular race-track is . The track is everywhere
wide. Calculate the cost of leveling the track at the rate of
paise per square meter.
Take
Answer:
Let the outer radius
Circumference
Therefore inner radius
Therefore area of track
Cost of leveling
Question 26: A rectangular piece is long and
wide. From its four corners, quadrants of radii
have been cut. Find the area of the remaining part.
Answer:
Area of the rectangle
Area of quadrants
Therefore Remaining area
Question 27: Four equal circles, each of radius , touch each other as shown in the adjoining figures. Find the area included between them.
Take
Answer:
Area of the square
Area of quadrants
Therefore Remaining area
Question 28: Four cows are tethered at four corners of a square plot of side , so that they just cannot reach one another. what area will be left ungrazed?
Answer:
Area of the square
Area of quadrants
Therefore Remaining area
Question 29: A road which is wide surrounds a circular park whose circumference is
. Find the area of the road.
Answer:
Let the radius of the park
Circumference of park
Therefore
Outer radius
Therefore area of the road
Question 30: Four equal circles, each of radius a, touch each other. show that the area between the is .
Take
Answer:
Area of rectangle = (2a) \times (2a) = 4a^2
Area of quadrants
Therefore Remaining area
Question 31: Two circular pieces of equal radii and maximum area, touching each other are cut out from a rectangular card board of dimensions . Find the area of the remaining cardboard.
Take
Answer:
Area of board
Area of two cut outs
Remaining area
Question 32: In the adjoining figure, a square is inscribed in a quadrant
of a circle. If
, find the area of the shaded region.
Answer:
Given: ,
is a square
Therefore Radius
Hence area of quadrant
Area of square
Therefore shaded area
Question 33: In the adjoining figure, is a right angled triangle in which
and
. Semi-circles are described on
and
as diameters. Find the area of the shaded region.
Answer:
Area of small semi circle with diameter
Area of large semi circle with diameter
Area of large semi circle with diameter
Area of
Therefore shaded area
Question 34: In the adjoining figure, and
is mid-point of
. Semi-circles are drawn on
and
as diameters. A circle with center
touches all the three circles. Find the area of the shaded region.
Answer:
Total area of large semi circle
Area of two smaller semi circles
Let the radius of the small circle
Therefore,
Therefore area of small circle
hence the shaded area
Question 35: In the adjoining figure, the boundary of the shaded region consists of four semi-circular arcs, the smallest two being equal. If the diameter of the largest is and of the smallest is
, find i) the length of the boundary ii) the area of the shaded region.
Answer:
Length of boundary
Therefore shaded area
Question 36: In the adjoining figure, is the center of a circular arc and
is a straight line. Find the perimeter and the area of the shaded region correct to one decimal place.
Take
Answer:
Therefore Radius
Perimeter
Shaded region
Question 37: In the adjoining figure, there are three semicircles, and
having diameter
each, and another semicircle
having a circle
with diameter
are shown. Calculate: (i) the area of the shaded region (ii) the cost of painting the shaded region at the rate of
, to the nearest rupee.
Answer:
i) Area of shaded area
ii) Therefore cost of painting shaded area
Question 38: In the adjoining figure and
are two diameters of a circle perpendicular to each other and
is the diameter of the smaller circle. lf
, find the area of the shaded region.
Answer:
Area of larger circle
Area of smaller circle
Therefore shaded region
Question 39: In the adjoining figure, is a quadrant of a circle with center
and radius
. If
is
, find the area of the i) quadrant
an ii) shaded region.
Answer:
i) Area of quadrant
ii) Area of shaded region
Question 40: For each of the two opposite corners of a square of side , a quadrant of a circle of radius
is cut. Another circle of radius
is also cut from the center as shown in the figure. Find the area of the remaining shaded portion of the square.
Take
Answer:
Shaded area
Question 41: Find the area of the shaded region in the adjoining figure, if , and
is center of the circle.
Take
Answer:
Therefore radius of the circle
Hence the shaded area
Question 42: In the adjoining figure, is a square of side
. If
is a quadrant of a circle with center
, then find the area of the shaded region.
Take
Answer:
Shaded area
Question 43: In the adjoining figure, is a rectangle, having
and
. Two sectors of
have been cut off. Calculate: i) the area of the shaded region ii) the length of the boundary of the shaded region.
Answer:
Area of the shaded region
Perimeter
Question 44: A circle is inscribed in an equilateral triangle is side
, touching its Sides as shown in the adjoining figure. Find the radius of the inscribed circle and the area of the shaded part.
Answer:
Area of
Let the radius of the circle
Therefore
Therefore area of circle
Therefore shaded area
Question 45: In the adjoining figure, shows the cross-section of railway tunnel. The radius of the circular part is
. If
, calculate: (i) the height of the tunnel (ii) the perimeter of the cross-section (iii) the area of the cross-section.
Answer:
i) Height of tunnel
ii) Perimeter
iii) Area of cross-section