Convex Polygon:
VII. Circular Pathway
OAC is a circle of radius = r, there is pathway, outside the circle of width = W Area of circular pathway = π W (2r+W)
When, the pathway is inside the circle, Area of circular pathway = πW (2r - W)
SOLVED EXAMPLES
1. If three sides of a triangle are 5, 6 and 7 cm respectively, find the area of triangle.
Sol: Area of = s(s a)(s b)(s c)
Now, s = a b c 5 6 7
2 2
= 9
Area = 9 (9 5)(9 6)(9 7) 9 4 3 2
= 216 6 6 cm2.
2. ABC is an equilateral triangle of side 24 cm. Find the in radius of the triangle.
Sol: In a equilateral triangle, the altitude, median and perpendicular are equal.
AD = 3 /2 x 24 = 12 3
GD (in radius) = 1/3 x 12 3 = 4 3 cm
3. The base and other side of an isosceles triangle is 10 and 13 cm respectively. Find its area.
Sol: Area of Isosceles = b 2 2 4a b
4
Given, base b = 10 Other side a = 13
Area (A) = 10 2 2 10
4 (13) 10 676 100
4 4
= 10
4 24 = 60 cm2.
4. In a right-angled triangle, the length of two legs are 12 and 5 cm. Find the length of hypotenuse and its area.
Sol: In a right angled triangle,
(Hypotenuse)2 = (one leg)2 + (other leg)2
= 122 + 52
Hypotenuse = 12252 = 169 = 13 cm.
In a right angled triangle,
Area = 1 (leg)1 (leg)2 1 12 5
2 2 = 30 cm2.
5. If the perimeter and diagonal of a rectangle and 14 and 15 cm respectively. Find its area.
Sol: In a rectangle,
(Perimeter)2
4 = (diagonal)2 + 2 x Area ; (14)2
4 = (5)2 + 2 x Area
2 x Area = 196
4 - 25 Area = 49 25 2
= 12 cm2.
6. Find the length of the diagonal and the perimeter of a square plot if its area is 900 square metres.
Sol: In a square, A = d2 p2 2 16
(Diagonal)2 = 2 x Area = 900
Diagonal (d) = 2 900 30 2 = 42.42 metres (Perimeter)2 = 16 x Area = 16 x 900
Perimeter (P) = 16 900 = 120 metres.
7. A field in the shape of a rhombus has the distances between pairs of opposite vertices as 14 m and 48 m. What is the cost (in rupees) of fencing the field at Rs.20 per metre?
Sol: The diagonals are 14 m and 48 m
Sides of rhombus = 14 2 48 2 625
2 2
= 25
Perimeter of rhombus = 4 x 25 = 100 m.
Cost of fencing the field = 100 x 20 = Rs.2000
8. In a trapezium, the length of parallel sides are 20 and 25 metres respectively and the perpendicular distance between the parallel sides is 12 metres. Find the area of trapezium.
Sol: One parallel side a = 20 metres. Second parallel side b = 25 metres. Height (perpendicular distance between a and b) = 12 metres.
Area = 1 1
(a b) h (20 25) 12
2 2 = 270 m2.
9. The distance between a pair of opposite vertices of a quadrilateral is 32 units. The lengths of the perpendiculars drawn on to this diagonal from the other two vertices are 4 1/3 units and 6 2/3 units respectively. Find the area (in sq units) of the quadrilateral?
Sol: Area of quadrilateral = 1/2 x 32 x 13 20
3 3
= 178 sq units.
10.
In the above parallelogram ABCD, A = x + 30o and D = x – 40o, what is the measure of DCB ?
Sol: In a parallelogram, sum of adjacent angles is equal to 180o
x + 30 + x – 40 = 180 x = 95o
DAB = x + 30 = 95 + 30 = 125o
DCB = DAB = 125o
(opposite angles of a parallelogram are equal)
A B
C D
11. In a circle of radius 49 cm, an arc subtends an angle of 36o at the centre. Find the length of the arc and the area of the sector.
Sol: Length of the arc = 2 r 2 22 49 36
360 7 360
θ
= 30.8 cm
Area of the sector = r2 22 49 49 36
360 7 360
θ
= 754.6 cm2
12. A rectangular plot of dimensions 13 m x 17 m is surrounded by a garden of w idth 5 m.
What is the area (in sq m) the garden?
Sol: Let ABCD be the rectangular plot of given dimension. The shaded part is the surrounding garden. Now, the plot ABCD together with the garden forms another rectangular form PQRS. Dimensions of PQRS, as can be seen from the diagram, are:
Length PQ = width of garden + AB + width of garden
= 5 + 17 + 5 = 27 m
Similarly, breadth = PS = 5 + 13 + 5 = 23 m Area of garden = Area of PQRS – Area of ABCD
= (27 x 23) – (17 x 13) = 621 – 221 = 440 sq m.
13. There is a rectangular field of length 100 m and breadth 40 m. A carpet of 2 m width is to be spread from the centre of each side to the opposite side. What is the area of the carpet?
Sol: Area of the carpet ABCD = 40 m x 2 m = 80 m2 Area of the carpet EFGH = 100 m x 2 m = 200 m2 But the common area of two carpets = 2 x 2 = 4m2
So, area of the carpet = 200 + 80 – 4 = 276 m2
14. There is an equilateral triangle of which each side is 3 m. With all the three vertices as centres, circles with radius 1.5 cm are described (i) Calculate the area common to all the circles and the triangle. (ii) Find the area of the remaining portion of the triangle.
Sol: (i) Area of each sector = 1 r2 6
So area common to the all the circles and triangle = 3 1 r2 1 r2
6 2
= 1 22 1.5 1.5
2 7 = 3.53 m2
(ii) Area of the shaded portion = Area of the triangle – Area common to the triangle and the circles
But area of the triangle = 3 2 3 2 9 3
a (3)
4 4 4 m2
So area of the shaded portion = 9 3
4 m2 – 3.53 m2 = 3.89 m2 – 3.53 m2 = 0.36 m2
Brainstorming
1. Find the area of a triangle having sides 3 m, 4 m and 5 m.
1. 60 sq m 2. 10 sq m 3. 12 sq m 4. 6 sq m 2. Find the area of a triangle whose base is 4.6m and height is 67 cm.
1. 154.10 sq m 2. 15410 sq m 3. 15.410 sq m 4. None 3. Find the area of an equilateral triangle each of whose sides measures 6 cm.
1. 36 sq cm 2. 3 3 sq cm 3. 9 3 sq cm 4. 12 sq cm 4. Length of the side of an equilateral triangle is
3
4 cm. Find its height.
1. 2 cm 2. 4 cm 3. 6 cm 4. None
5. Height of an equilateral triangle is 4 3 cm. Find its area.
1. 4 3sq cm 2. 2 3sq cm 3. 16 3sq cm 4. 8 3sq cm 6. An isosceles right-angled triangle has two equal sides of length 6 m each. Find its area
1. 8 sq m 2. 36 sq m 3. 18 sq m 4. None
7. The perimeter of an isosceles triangle is 80 cm. If the length of the equal sides is given by 0.15 m, find the length of the base.
1. 40 m 2. 50 m 3. 12 m 4. 90.5 m
8. The perimeter of an isosceles triangle is 42 cm. If the base is 16 cm, find the length of equal sides.
1. 13 cm 2. 8 cm 3. 21 cm 4. 29 cm
9. The two adjacent sides of a parallelogram are 5 m and 6 m respectively, and if the diagonal connecting the ends is 9 m, find the area of the parallelogram (approximately).
1. 29 sq m 2. 28 sq m 3. 58 sq m 4. 50 sq m
10. Find the area of a quadrilateral of whose diagonal is 38 cm long and the lengths of perpendiculars from the other two vertices are 31 cm and 19 cm, respectively.
1. 950 sq cm 2. 475 sq cm 3. 138 sq cm 4. 276 sq cm
11. Find the area of a parallelogram whose two adjacent sides are 130 m and 140 m and one of the diagonals is 150 m long.
1. 8400 sq cm 2. 16,800 sq cm 3. 2100 sq cm 4. None 12. Find the diagonal of a rectangle whose sides are 8 cm and 6 cm.
1. 14 cm 2. 5 cm 3. 20 cm 4. 10 cm
13. Find the perimeter of a rectangle of length 12 m and breadth 6 m.
1. 18 m 2. 72 m 3. 36 m 4. 144 m
14. Calculate the area of a rectangular field whose length is 12.5 cm and breadth is 8 cm.
1. 10 sq cm 2. 100 sq cm 3. 200 sq cm 4. 1 sq cm
15. Calculate the area of a rectangular field whose one side is 16 cm and the diagonal is 20 cm.
1. 192 sq cm 2. 96 sq cm 3. 294 sq cm 4. 72 sq cm
16. A rectangular carpet has an area of 120 sq m and perimeter of 46 m. Find the length of its diagonal.
1. 34 m 2. 51 m 3. 93 m 4. 17 m
17. The perimeter of a rectangle is 82 cm and its area is 400 sq m. Find the length of the rectangle.
1. 8 m 2. 16 m 3. 32 m 4. 64 m
18. If the area of a square field be 6050 sq m, find the length of its diagonal.
1. 220 m 2. 110 m 3. 55 m 4. None
19. Find the area of a square with perimeter 48 m.
1. 288 sq m 2. 72 sq m 3. 144 sq m 4. 96 sq m 20. Find the diagonal of a square field whose side is of 6 m length.
1. 12 2 m 2. 6 2 m 3. 2 m 4. 3 2 m
21. Perimeter of a square field is 16 2cm. Find the length of its diagonal.
1. 16 cm 2. 4 cm 3. 8 cm 4. 64 cm
22. The area of a rhombus is 156 sq m. If one of its diagonals is 13 m, find the length of the other diagonal.
1. 12 m 2. 6 m 3. 48 m 4. 24 m
23. Find the area of a rhombus whose one side is 13 cm and one diagonal is 24 cm.
1. 60 sq cm 2. 120 sq cm 3. 240 sq cm 4. 74 sq cm 24. If the perimeter of a rhombus is 73 cm and one of its diagonals is 27.5 cm, find the other
diagonal and the area of the rhombus.
1. 24 cm, 330 sq cm 2. 20 cm, 115 sq cm 3. 30 cm, 660.8 sq cm 4. 40 cm, 100.5 sq cm 25. In a rhombus, the lengths of two diagonals are 18 m and 24 m. Find its perimeter.
1. 15 m 2. 30 m 3. 60 m 4. 120 m
26. The diagonally of Rhombus are 12 cm and 5 cm respectively. Find the side of the Rhombus.
1. 5 cm 2. 6.5 cm 3. 6 cm 4. 8.5 cm
27. What is the radius of a circular plot whose circumference is 176 m?
1. 14 m 2. 56 m 3. 88 m 4. 28 m
28. A circular plot covers an area of 154 sq m. How much wire is required for fencing the plot?
1. 44 m 2. 22 m 3. 88 m 4. 77 m
29. Find the area of sector of a circle whose radius is 10 cm and the angle at the center is 36o. 1. 30
7
3 sq cm 2. 31 3
7 sq cm 3. 30 3
7 sq cm 4. 31 7
3 sq cm 30. Find the area of sector of a circle whose radius is 12 cm and the length of the arc is 20 cm.
1. 60 sq cm 2. 240 sq cm 3. 120 sq cm 4. 64 sq cm
31. Find the side of a regular hexagon whole area is 30 3 sq cm.
1. 5 cm 2. 2 5 cm 3. 3 5 cm 4. None
32. Find the area of a regular octagon whole side measures 8 cm.
1. 4( 2 +1) sq cm 2. 8( 2 +1) sq cm 3. 16 ( 2 +1) sq cm 4. ( 2 +1) sq cm 33. Find the sum of interior angles of a regular polygon of 12 sides. Also, find the value of each
interior angle.
1. 10 Π , 6
Π
5 2. ∏,
6 Π
5 3. 8 Π ,
5 Π
6 4. None
34. Find the sum of all the exterior angles of a regular polygon of 10 sides. Also, find the value of each exterior angle.
1. Π , 5
Π 2. 2 Π ,
5
Π 3. 3 Π ,
5
Π 4. None
35. The length and breadth of a rectangle are increased by 20% and 5%, respectively. Find the percentage increase in its area.
1. 25% 2. 26% 3. 13% 4. 15%
36. Two poles 15 m and 30 m high stand up right in a play ground if their feet be 36 cm a part find the distance between their tops.
1. 41 cm 2. 39 cm 3. 29 cm 4. 42 cm
37. If all the sides and the diagonals of a square are increased by 8% each, then find the percentage increase in its perimeter?
1. 8% 2. 6% 3. 1% 4. None
38. Ratio of the areas of two squares is 16 : 9. Find the ratio of their diagonals.
1. 2 : 4 2. 9 : 16 3. 4 : 3 4. 64 : 9
39. The diagonal of a square is doubled. How many times will the area of the new square become?
1. 2 times 2. 4 times 3. 6 times 4. 8 times
40. How many meters of a carpet 12 cm wide will be required to cover the floor of a room which is 600 cm long and 420 cm broad? Also, calculate the amount required in carpeting the floor if the cost of carpet is Rs.15 per meter.
1. Rs.3150 2. Rs.9000 3. Rs.1800 4. Rs.10,800
41. A hall of length 24 cm and breadth 20 cm is to be paved with equal square tiles. What will be the size of the largest tile so that the tiles exactly fit and also find the number of tiles required?
1. 60 2. 30 3. 480 4. 120
42. A rectangular park 18 m x 12 m, is surrounded by a path 4 m wide. Find the area of the path.
1. 304 sq m 2. 152 sq m 3. 608 sq m 4. 864 sq m
43. A park is square in shape with side 18 m. Find the area of the pavement 3 m wide to be laid all around it on its inside.
1. 360 sq m 2. 180 sq m 3. 90 sq m 4. None
44. A playground measures 27 m x 13 m. From the center of each side a path 2 m wide goes across to the center of the opposite side. Calculate the area of the path and the cost of constructing it at Rs.4 per sq m.
1. Rs.101 2. Rs.404 3. Rs.202 4. Rs.304
45. A square field is surrounded by a path 2 m wide on its outside. The area of the path is 72 sq m.
What is the area of the field?
1. 15 sq cm 2. 121 sq m 3. 36 sq m 4. 18 sq m
46. A circular park of radius 22 m has a path of width 1.4 m around it on its inside. Find the area of the path.
1. 178.45 sq m 2. 187.45 sq m 3. 187.54 sq m 4. None
47. If the area of a square is 33 sq cm, then find the area of the circle formed by the same perimeter.
1. 21 sq cm 2. 66 sq cm 3. 33 sq cm 4. 42 sq cm
48. Find the area of largest circle inscribed in a square of side 112 cm.
1. 4928 sq cm 2. 8856 sq cm 3. 9856 sq cm 4. None 49. Find the side of the square inscribed in a circle whose circumference is 308 cm.
1. 25 2 cm 2. 49 2 cm 3. 36 2 cm 4. 16 2 cm
50. The diameter of a wheel is 2 cm. If it rolls forward covering 10 revolutions, find the distance traveled by it.
1. 62.8 cm 2. 31.4 cm 3. 15.7 cm 4. 58.2 cm
1) 4 11) 2 21) 3 31) 2 41) 4
2) 2 12) 4 22) 4 32) 3 42) 1
3) 3 13) 3 23) 1 33) 1 43) 2
4) 1 14) 2 24) 1 34) 2 44) 4
5) 3 15) 1 25) 3 35) 2 45) 2
6) 3 16) 4 26) 2 36) 2 46) 2
7) 2 17) 2 27) 4 37) 1 47) 4
8) 1 18) 2 28) 1 38) 3 48) 3
9) 2 19) 3 29) 4 39) 2 49) 2
10) 1 20) 2 30) 3 40) 1 50) 1
21. MENSURATION : 3D
Cuboid :
A right prism with a rectangular base is called a Cuboid.
The sides of the base are length (l) and breadth (b). The height is h.
Lateral Surface Area = 2h(l + b)
Total Surface Area = 2h(l + b) + 2lb = 2(lb + bh + hl) Longest diagonal = l2b2h2
Volume = lbh
Cube:
If the length, breadth and height of a cuboid are all equal, it is called a cube.
Then, if edge of the cube = a Longest diagonal = 3 a Lateral Surface Area = 6a2 Total surface Area = 6a2
Volume = a3
Cylinder :
A cylinder can be considered to be a right prism except that instead of identical polygons a cylinder has identical circles for its top and base and it has a single lateral surface also called curved surface, instead of several rectangular surfaces.
The basic measurements are the radius of the base (or top) r and the height h.
Curved Surface (Lateral Surface Area) = 2π rh
Total surface Area = 2π rh+2 π r2 = 2π r(h + r) Volume = π r2h