Geometry and Mensuration  Aptitude basics, practice questions, answers and explanations 
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Geometry and Mensuration

Mensuration: Mensuration is the branch of mathematics which deals with the study of Geometric shapes , Their area , Volume and different parameters in geometric objects.

Some important mensuration formulas are:

1.Area of rectangle (A) = length(l) * Breath(b);

2.Perimeter of a rectangle (P) = 2 * (Length(l) + Breath (b))

3.Area of a square (A) = Length (l) * Length (l)

4.Perimeter of a square (P) = 4 * Length (l)

5.Area of a parallelogram(A) = Length(l) * Height(h)

6.Perimeter of a parallelogram (P) = 2 * (length(l) + Breadth(b))

7.Area of a triangle (A) = (Base(b) * Height(b)) / 2
  

And for a triangle with sides measuring “a” , “b” and “c” , Perimeter = a+b+c

and s = semi perimeter = perimeter / 2 = (a+b+c)/2

And also . Area of triangle,

  

This formulas is also knows as “Hero’s formula”.

8.Area of triangle(A)
  = 

9. Area of isosceles triangle = 

Where , a = length of two equal side , b= length of base of isosceles triangle.

10.Area of trapezium (A) =(a+b)/2

Where , “a” and “b” are the length of parallel sides.

11.Perimeter of a trapezium (P) = sum of all sides

12.Area f rhombus (A) = Product of diagonals / 2

13.Perimeter of a rhombus (P) = 4 * l

where l = length of a side

14.Area of quadrilateral (A) = 1/2 * Diagonal * (Sum of offsets)

15. Area of a Kite (A) = 1/2 * product of it’s diagonals

16. Perimeter of a Kite (A) = 2 * Sum on non-adjacent sides

17. Area of a Circle (A)  =

  , where , r= radius of the circle

18. Circumference of a Circle  =

r= radius of circle

d= diameter of circle

19. Total surface area of

cuboid = 2(lb+bh+lh)

where , l= length , b=breadth , h=height

20. Total surface area of

cuboid =  6l²

where , l= length

21. length of diagonal of cuboid =

22. length of diagonal of cube = 

23. Volume of cuboid = l * b * h

24. Volume of cube = l * l* l

25. Area of base of a cone =

26. Curved surface area of a cone =C

Where , r = radius of base , l = slanting height of cone

27. Total surface area of a cone = 

28. Volume of right circular cone = 

Where , r = radius of base of cone , h= height of the cone (perpendicular to base)

29. Surface area of triangular prism = (P * height) + (2 * area of triangle)

Where , p = perimeter of base

30. Surface area of polygonal prism = (Perimeter of base * height ) + (Area of polygonal
base * 2)

31. Lateral surface area of prism = Perimeter of base * height

32. Volume of Triangular prism = Area of the triangular base * height

33. Curved surface area of a cylinder =

Where , r = radius of base , h = height of cylinder

34. Total surface area of a cylinder = 

35. Volume of a cylinder =

36. Surface area of sphere = 

where , r= radius of sphere , d= diameter of sphere

37. Volume of a sphere =

38. Volume of hollow cylinder =

where , R = radius of cylinder , r= radius of hollow , h = height of cylinder

39. Surface area of a right square pyramid =

Where , a = length of base , b= length of equal side ;

of the isosceles triangle forming the slanting face.

40. Volume of a right square pyramid =

41. Area of a regular hexagon =

42. area of equilateral triangle = 

43. Curved surface area of a Frustums = 

44. Total surface area of a Frustums =  

45. Curved surface area of a Hemisphere = 

46. Total surface area of a Hemisphere =

47. Volume of a Hemisphere = 

48. Area of sector of a circle = 

where , \theta = measure of angle of the sector , r= radius of the sector

Exercise questions

1. What is the area of an equilateral triangle of side 16cm ?
a) 243 cm2
b) 64 3 cm2
c) 363 cm2
d)323 cm2

2. Consider the following figure. <A=x+30; <D= x-40. Find <B?
a. 125o
b. 55o
c. 155o
d. 122o

3. Find the area of a square, the product of whose diagonals is 66 cm2
a) 30 cm2
b) 33 cm2
c) 36 cm2
d) 42 cm2

4. A 5 cubic centimeter cube is painted on all its side. If it is sliced into 1 cubic centimeter cubes, how many 1 cubic centimeter cubes will have exactly one of their sides painted?
a) 9
b) 61
c) 98
d) 54

5. Find the area of a trapezium whose parallel sides are 20 cm and 18 cm long, and the distance between them is 15 cm.
a) 225 cm2
b)275 cm2
c) 285 cm2
d)315 cm2

6. Examine the figure. <ADB=25o; Find <OBC:
a. 115o
b. 25o
c. 50o
d. 65o

7. The sector of a circle has radius of 21 cm and central angle 1350. Find its perimeter.
a) 91.5 cm
b)93.5 cm
c) 94.5 cm
d) 92.5 cm

8. The volumes of two cones are in the ratio of 1 :10 and the radii of the cones are in the ratio of 1 : 2, what is the ratio of their vertical heights?
a) 2 : 5
b) 1 : 5
c) 3 : 5
4) 4 : 5

Answer Key

1.c; 2.b; 3.b; 4.d; 5.c; 6.a; 7.a; 8.a