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