Question 1: A bucket is raised from a well by a means of a rope which is wound around a wheel of diameter . Given that the bucket ascents in 1 minute 28 seconds with a uniform speed of , calculate the number of complete revolution the wheel will make in raising the bucket. [1997]

Answer:

Radius of the wheel

Length of the rope

Circumference of the wheel

Therefore No of revolutions

Question 2: The wheel of a cart is making revolutions per second. If the diameter of the wheel is , find the speed in . Give your answer, correct to the nearest km. [1998]

Answer:

No of revolutions

Radius of the wheel

Speed

Question 3: In the given figure. is the diameter of the circle with center and . Find the area of the shaded region. [2006]

Answer:

and

Area of smaller circle

Area of the triangle

Area of semi-circle

Therefore the area of the shaded area

Question 4: and are two mutually perpendicular diameters of a circle . Given the area of the shaded region is Calculate the (i) the length of and (ii) the circumference of the circle [Take ] [2009]

Answer:

Given Area of the shaded region :

(i) Therefore

(ii) Circumference of the circle

Question 5: A doorway is decorated as shown in the figure. There are four semi-circles. , the diameter of the larger Semi-circles. the diameter of the larger semi-circle is of length . Center of the three equal semi-circles lie on . is an isosceles triangle with . If , find the area of the shaded, region. [2010]

Answer:

Let

As angle in semi-circle is

In right angled , by Pythagoras theorem, we get

Also, Area of

Diameter of semicircle

Area of semi-circle

Diameter of each (three equal) semi-circles

Radius of the 3 equal semi-circles

Therefore Area of three equal semi circles

Area of shaded region = Area of semi-circles + Area of three equal circles – Area of

Question 6: The shaded part of the given figure shows the shape of the top of a table in a restaurant which is a segment of a circle with center , and . Find (i) the area of the top of the table (ii) the perimeter of the table (). [2002]

Answer:

Area of the table

Perimeter of the table

Question 7: Calculate the area of the shaded portion. The quadrants shown in the figure are each of radius . [2000]

Answer:

Area of the shaded region

Question 8: In the figure given below is a rectangle . From the rectangle a quarter circle and a semicircle are removed Calculate the area of the remaining piece of the rectangle (Take ) [2014]

Answer:

Area of rectangle

Area of quarter circle

Area of semicircle

Area of remaining piece of rectangle

Question 9: The given figure shows a running track surrounding a grass enclosure . The enclosure consists of a rectangle with a semicircular region at each end. and . (i) calculate the area of the grass enclosure in (ii) given that the track is of constant width , calculate the outer perimeter of the track. [1999]

Answer:

Radius of the small semi-circle

Radius of the large semi-circle

(i) Area of grass

(ii) Outer perimeter

Question 10: A rectangular playground has two semi circles added to the outside with its smaller sides as diameters. If the sides of the rectangle are and , find the area of the playground (). [2000]

Answer:

Area of the playground

Question 11: In the figure along side is a quadrant of a circle, The radius , Calculate the area of the shaded portion. (Take ) [2013]

Answer:

Radius of quadrant

Area of quadrant

Here,

and

Then area of

Area of shaded portion =Area of quadrant – Area of triangle

Question 12: is an is isosceles right-angled triangle with . A semi-circle is drawn with as the diameter. If find the area of the shaded region. . [2012]

Answer:

is a right angled triangle. Therefore

Area of semi circle

Area of

Area of the shaded region = Area of the semi circle – Area of

Question 13: (i) From a rectangular card board , two circles and one semi-circle of the largest sizes are cut out as shown below. Calculate the ratio of the area of the remaining card board and the area of the card board cut.

(ii) If the figure of part (i), given above, find the area of the shaded portion within the rectangle, if radius of each circle is and . [2008]

Answer:

Let the radius of the circle

Therefore and

Area of rectangle

Area of the cardboard cut out

Area of remaining cardboard

(i) Ratio

(ii) Area of the shaded region

Question 14: In an equilateral of side , side is the diameter of a semi-circle as shown in the given figure. Find the area of the shaded region. [Take and ] [2007]

Answer:

Area of the semi-circle

Height of the triangle

Area of triangle

Therefore the area of the shaded region

Question 15: Calculate the area of the shaded region, if the diameter of the semi circle is equal to . (Take ) [2011]

Answer:

Area of shaded portion = Total area – area of the two quadrants